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In 1962, Dyson showed that the spectrum of a random Hermitian matrix, whose
entries (real and imaginary) diffuse according to independent Ornstein-Uhlenbeck
processes, evolves as non-colliding Brownian particles held together by a drift term.
When , the largest eigenvalue, with time and space properly
rescaled, tends to the so-called Airy process, which is a non-markovian continuous
stationary process. Similarly the eigenvalues in the bulk, with a different time and
space rescaling, tend...
We show that the periodic Camassa–Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data in . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure with . The total energy is preserved by the solution.
A new, shorter, proof of the Treves theorem on an algebraic criterion for the first
integrals of the KdV hierarchy is given, along with an addition to the theorem.
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