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The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1] [2]In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is
The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables. Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution.
The paradigmatic case is the stability of the origin under the linear autonomous differential equation ˙ = where = [] and is a 2-by-2 matrix. We would sometimes perform change-of-basis by X ′ = C X {\displaystyle X'=CX} for some invertible matrix C {\displaystyle C} , which gives X ˙ ′ = C − 1 A C X ′ {\displaystyle {\dot {X}}'=C^{-1 ...
For a rational and continuous-time system, the condition for stability is that the region of convergence (ROC) of the Laplace transform includes the imaginary axis.When the system is causal, the ROC is the open region to the right of a vertical line whose abscissa is the real part of the "largest pole", or the pole that has the greatest real part of any pole in the system.
A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is determined by applying any one of several stability criteria.
Two beta-decay stable nuclides exist for odd neutron numbers 1 (2 H and 3 He), 3 (5 He and 6 Li – the former has an extremely short half-life), 5 (9 Be and 10 B), 7 (13 C and 14 N), 55 (97 Mo and 99 Ru), and 85 (145 Nd and 147 Sm); the first four cases involve very light nuclides where odd-odd nuclides are more stable than their surrounding ...
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The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. [ 1 ] [ 2 ] Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel).