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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 ...
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
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.
Computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x 0 to , for instance x 0 = 1.4, and then computing improved guesses x 1, x 2, etc. One such method is the famous Babylonian method, which is given by x k+1 = (x k + 2/x k)/2.
In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. [1] The distributions of random variables having this property are said to be "stable distributions".
Four of those pillars of stability are the West Antarctic Ice Sheet, the Greenland Ice Sheet, and Amazon Rainforest, and the Atlantic Meridional Overturning Circulation (AMOC)—the ocean current ...
Let X 1 and X 2 be independent realizations of a random variable X. Then X is said to be stable if for any constants a > 0 and b > 0 the random variable aX 1 + bX 2 has the same distribution as cX + d for some constants c > 0 and d. The distribution is said to be strictly stable if this holds with d = 0. [7]
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