ergodic theorem in a sentence

Examples

1. Note that, while formally the connection between such averages is justified in equilibrium statistical mechanics by the ergodic theorem, the statistical mechanics of hydrodynamic turbulence is currently far from understood.
2. Remark : Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane ( by left multiplication ).
3. Doob's approach to probability was evident in his first probability paper, in which he proved theorems related to the law of large numbers, using a probabilistic interpretation of Birkhoff's ergodic theorem.
4. In mathematics, the "'multiplicative ergodic theorem "', or "'Oseledets theorem "'provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system.
5. It was U . Krengel who first gave, in 1971, an ingenious construction of an increasing sequence of positive integers along which the pointwise ergodic theorem fails in " L " 1 for every ergodic transformation.
6. Beck graduated from Stuyvesant High School in 1947, studied at Brooklyn College ( Bachelor's degree 1951 ) and in 1956 received his PhD from Yale University under Shizuo Kakutani PhD ( On the Random Ergodic theorem ).
7. A few years later, R . V . Chac�n gave an example of a positive ( linear ) isometry of " L " 1 for which the individual ergodic theorem fails in " L " 1.
8. That both of these computations give the same stationary vector is a form of an ergodic theorem, which is generally true in a wide variety of dissipative dynamical systems : the system evolves, over time, to a stationary state.
9. More precise information is provided by various "'ergodic theorems "'which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average.
10. The theorem was proven in 1923 by G . D . Birkhoff ( author of an even more famous " Birkhoff theorem ", the " pointwise ergodic theorem " which lies at the foundation of ergodic theory ).