[Cryptographic] properties of rule 30 Rule 30 can be written in the form p ⊻ (q ∨ r) (see page 869 ) and thus exhibits a kind of one-sided additivity on the left. … It turns out that this can be done even if the right-hand one of the two adjacent columns is not complete. … Many distinct columns correspond to starting at different points on a single cycle of states.

Various exact results were obtained—notably the existence of stable equilateral triangle configurations corresponding to so-called Lagrange points. … With appropriate initial conditions one can get various forms of simple behavior. … What generically happens is that one of the bodies escapes from the other two (like t or sometimes t 2/3 ).

But if one allows progressively more molecules computational irreducibility can make it take progressively more computational work to see what will happen. … Thus in practice it has typically been difficult to predict for example boiling and particularly melting points (see note below ).

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

In mathematical terms, gauge theories can be viewed as describing fiber bundles in which connections between values of group elements in fibers at neighboring spacetime points are specified by gauge potentials—and curvatures correspond to gauge fields.

Detailed ideas about infinite sets emerged in the 1880s through the work of Georg Cantor , who found it useful in studying trigonometric series to define sets of transfinite numbers of points. … As discussed on page 1176 , however, one cannot avoid axiom schemas in the formulation of set theory given here.

But particularly with the development of game theory in the 1940s the notion became established, at least in theoretical economics, that prices represent equilibrium points whose properties can be derived mathematically from requirements of optimality.

At first it was thought that the visual system might be sensitive only to the overall autocorrelation of an image, given by the probability that randomly selected points have the same color. … One was largely an outgrowth of work in artificial intelligence, and concentrated mostly on trying to use traditional mathematics to characterize fairly high-level perception of objects and their geometrical properties.

To investigate collisions between particles, one thus looks at what happens with multiple waves. … (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.) … One of the apparent implications of QCD is the confinement of quarks and gluons inside color-neutral hadrons.

He argued that if the time evolution of a single state were to visit all other states in the ensemble—the so-called ergodic hypothesis—then averaged over a sufficiently long time a single state would behave in a way that was typical of the ensemble.