Topological defects An idealized vortex in a 2D fluid involves velocity vectors that in effect wind around a point—and can never be unwound by making a series of small local perturbations.

In quantum field theory and to a lesser extent quantum mechanics and celestial mechanics, approximation schemes based on perturbation series seem to require computations that grow very rapidly with order.

As a specific example consider taking a series of steps, and at each step picking a square in the array discussed above at random, then reversing the color of this square whenever doing so will not increase the total number of squares in the array that violate the constraints.

And in the past few sections [ 10 , 11 , 12 ] I have outlined a series of types of models for how such a causal network might actually get built up.

Connections to formal power series and to substitution systems (see page 891 ) were studied in the 1960s.

Each theorem defines a connection between strings, and proving the theorem consists in finding a series of transformations—each associated with an axiom—that lead from one string to another.

Early mythologies most often assume that the universe started in chaos, with a supernatural being adding order, then creating a series of specific complex natural systems. … Typical was John Ray 's 1691 book The Wisdom of God Manifested in the Works of the Creation, which gave a long series of examples from biology that it claimed were so complex that they must be the work of a supernatural being.

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. … (For rational f Paul Painlevé in the 1890s identified just 6 additional types of functions that are needed, but even now series expansions are not known for all of them.)

The most obvious iterative procedure to use for such a problem involves taking a series of small steps, with the direction of each step being chosen so as locally to go downhill.

But eventually there will normally be places where the pattern is not yet uniquely determined, and so a series of choices have to be made.