Editing Singularity (mathematics) (section)

==Finite-time singularity==
[[File:Rectangular hyperbola.svg|thumb|The [[reciprocal function]], exhibiting [[hyperbolic growth]].]]<!-- A better image would be 1/(1-x) or similar, showing a positive singular point and growth as x increases -->

A '''finite-time singularity''' occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in [[kinematic]]s and [[Partial Differential Equation]]s – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are [[power law]]s for various exponents of the form <math>x^{-\alpha},</math> of which the simplest is [[hyperbolic growth]], where the exponent is (negative) 1: <math>x^{-1}.</math> More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses <math>(t_0-t)^{-\alpha}</math> (using ''t'' for time, reversing direction to <math>-t</math> so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of [[kinetic energy]] is lost on each bounce, the [[frequency]] of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the [[Painlevé paradox]] (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the [[precession]] rate of a [[coin]] spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the [[Euler's Disk]] toy).

Hypothetical examples include [[Heinz von Foerster]]'s facetious "[[Heinz von Foerster#Doomsday equation|Doomsday's equation]]" (simplistic models yield infinite human population in finite time).