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We show the Godbillon-Vey invariant arises as a `restricted Casimir invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of
We study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection. We assume that the three constants of motion are the Hamiltonian, $H$, with the standard form of a kinetic
We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operato
We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model, the Schonmann projection. These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function
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