Heuristic arguments are presented supporting the conjecture that almost all quantum Hamiltonians are integrable in the sense that there exist N ( N = n u m b e r of freedoms) mutually commuting observables (which, in analogy with the classical action variables, can be chosen to be the number operators). This follows from perturbational considerations: the series may converge for almost all perturbations that preserve the discreteness of the spectra, because a 'quantum small denominator' almost always uniformly satisfies the condition of sufficient irrationality. The radius of convergence vanishes if h = 0. The classical limit (as h + 0) of the quantum integrals of motion generically does not exist.

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The excitation of sound vibrations of a cylindrical fine fiber due to the impact of a spherical aerosol particle is investigated. The equations describing the dynamics of impact are derived for an arbitrary shooting parameter. The coefficient of restitution is calculated, and its analytical approximation is obtained. It is shown, for the case of long fibers, that the coefficient of restitution depends upon a single parameter λ(c). The parameter λ(c) depends on the particle radial velocity component near the fiber surface, the mass of the particle, the density of the fiber, the modulus of elasticity, and the geometric parameters of the fiber and the particle. The inertial deposition of submicron aerosol particles on fine fibers in a filter is considered. The efficiency of filtration is studied as a function of the gas flow velocity. The existence of a critical flow velocity U(*), below which the losses of particle energy during collision have no effect on the efficiency, is demonstrated. For velocities higher than the critical velocity, the filtration efficiency is dependent on the mechanisms of nonelastic losses of the particle's energy. Its value can be significantly lower than that estimated when particle rebound effects are neglected. After they have rebounded, some particles are not able to attain the initial high velocities in the stream, thus depositing on neighboring fibers. The dynamics of these particles is investigated. For this case, it is shown that the filtration efficiency is dependent on the velocity distribution of the rebounded particles and that it increases with the packing density of fibers. A qualitative difference between the asymptotic behavior of a fiber and that of a flat plate is found long after the initial impulse.

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