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.

The approach I discuss in the main text is quite different, in effect using the idea that in a network model of space there can be direct connections between particles that do not in a sense ever have to go through ordinary intermediate points in space. … And to get a kind of analog of Bell's inequalities one can look at correlations defined by such expectation values for field operators at spacelike-separated points (too close in time for light signals to get from one to another).

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.

Thus in practice it has typically been difficult to predict for example boiling and particularly melting points (see note below ).

Many distinct columns correspond to starting at different points on a single cycle of states.

• Is there a path shorter than some given length that visits all of some set of points in the plane?

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.

The first of these points means that one is concerned with so-called pure group theory—and with finding results valid for all possible groups.

Various exact results were obtained—notably the existence of stable equilateral triangle configurations corresponding to so-called Lagrange points.