Singularity (mathematics)
Finite-time singularity

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 kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form x − α, {\displaystyle x^{-\alpha },} of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x − 1. {\displaystyle x^{-1}.} More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses (t 0 − t) − α {\displaystyle (t_{0}-t)^{-\alpha }} (using t for time, reversing direction to − t {\displaystyle -t} so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time t 0 {\displaystyle t_{0}}).

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 " Doomsday's equation " (simplistic models yield infinite human population in finite time).
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