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Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence. In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such ...
Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots.
For example, if u 1 is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u 1, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point. There are also many other discrete dynamical systems.
A familiar example of parametric oscillation is "pumping" on a playground swing. [ 4 ] [ 5 ] [ 6 ] A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing ...
According to Nirenberg (1985, p. 703 and p. 707), [1] the space of functions of bounded mean oscillation was introduced by John (1961, pp. 410–411) in connection with his studies of mappings from a bounded set belonging to into and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following ...