Let A be an algebra over a field F of characteristic zero and let c n (A), n = 1, 2,. .. , be its sequence of codimensions. We prove that if c n (A) is exponentially bounded, its exponential growth can be any real number > 1. This is achieved by constructing, for any real number α > 1, an F-algebra A α such that lim n→∞ n √ c n (A α) exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.

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Abstract. We study Poisson algebras satisfying polynomial identities. In particular, such algebras satisfy "customary" identities (Farkas, 1998(Farkas, , 1999 Our main result is that the growth of the corresponding codimensions of a Poisson algebra with a nontrivial identity is exponential, with an integer exponent. We apply this result to prove that the tensor product of Poisson PI algebras is a PI-algebra. We also determine the growth of the Poisson-Grassmann algebra and of the Hamiltonian algebras H 2k .

The experimental results in a low-pressure r.f. self-sustained plasma discharge are discussed. A parametric coupling between several electronic modes with different azimuthal wavenumbers and an ionic mode are seen to explain one of the emissions whose frequency lies close to ionic plasma frequency. This oscillation is also observed when the exciting frequency lies close to the thermal Tonks-Dattner resonances (secondary resonances) frequencies in a low pressure glow discharge; in this case, by reducing the microwave pump power below the threshold required for the instability, a C.W. input signal whose frequency is in the vicinity of one of the parametric resonances, is amplified when this signal is excited with an azimuthal wavenumber differing from the exciting one.