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The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890.
Poincaré recurrence theorem, on sufficient conditions for recurrence to take place in dynamical systems; Poincaré-Bendixson theorem, on the existence of attractors for two-dimensional dynamical systems; Poincaré–Birkhoff–Witt theorem, concerning lie algebras and their universal envelopes; Poincaré lemma
In the context of metric measure spaces, the definition of a Poincaré inequality is slightly different.One definition is: a metric measure space supports a (q,p)-Poincare inequality for some , < if there are constants C and λ ≥ 1 so that for each ball B in the space, ‖ ‖ () ‖ ‖ ().
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount; Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution; Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS ...
The theorem is named after Henri Poincaré — who conjectured it in 1883 — and Carlo Miranda — who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. [ 1 ] [ 2 ] : 545 [ 3 ] It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.
Poincaré–Birkhoff–Witt theorem: an explicit description of the universal enveloping algebra of a Lie algebra. Poincaré–Bjerknes circulation theorem: theorem about a conservation of quantity for the rotating frame. Poincaré conjecture (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove ...
The theorem is named after Henri Poincaré. More specifically, let A be an n × n real symmetric matrix and B an n × r semi-orthogonal matrix such that B T B = I r. Denote by , i = 1, 2, ..., n and , i = 1, 2, ..., r the eigenvalues of A and B T AB, respectively (in descending order). We have