ergodic theorem वाक्य
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- 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.
- 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 ).
- 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.
- In mathematics, the "'multiplicative ergodic theorem "', or "'Oseledets theorem "'provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system.
- 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.
- 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 ).
- 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.
- 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.
- 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.
- 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 ).
- He is the author of the research monograph " Wiener Wintner Ergodic Theorems " ( World Scientific, 2003 ), about mathematics related to the Wiener Wintner theorem, and is also the editor of several volumes of collected papers.
- Birkhoff was influential in advancing the field of mathematics, solving Poincar?s " last geometric theorem ", and developing what is now called the " ergodic theorem ", a thesis important in statistical physics and the study of dynamic systems.
- More precisely, the "'pointwise "'or "'strong ergodic theorem "'states that the limit in the definition of the time average of ? exists for almost every " x " and that the ( almost everywhere defined ) limit function ? is integrable:
- In 1931 Koopman and Andr?Weil independently observed that the phase space of the classical system can be converted into a Hilbert space by postulating a natural integration rule over the points of the phase space as the definition of the scalar product, and that this transformation allows drawing of interesting conclusions about the evolution of physical observables from mean ergodic theorem ".
- This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff . ( Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold . ) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.
- This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff . ( Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold . ) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.
- One of the two problems that she raised at the Oberwolfach meeting on " Measure Theory " in 1981, was the question of the validity, for " ? " in " L " 1, of the pointwise ergodic theorem along the sequence of squares, and along the sequence of primes ( A similar question was raised independently, a year later, by H . Furstenberg ).
- His analysis of the structure of self-replication preceded the discovery of the structure of DNA . In a short list of facts about his life he submitted to the National Academy of Sciences, he stated " The part of my work I consider most essential is that on quantum mechanics, which developed in G�ttingen in 1926, and subsequently in Berlin in 1927 1929 . Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935 1939; on the ergodic theorem, Princeton, 1931 1932 ."
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