We present a set of quantum-mechanical Hamiltonians which can be written as the

Supersymmetric (pseudoclassical) mechanics has recently been generalized to fractional supersymmetric mechanics. In such a construction, the action is invariant under fractional supersymmetry transformations, which are the F th roots at time translations ( with F = 1, 2,…). Associated with these symmetries, there are conserved charges with fractional canonical dimension 1+1/F. Using paragrassmann variables satisfying θF=0, we present a fractional-superspace formulation of this construction.

We present a fractional superspace formulation of the centerless parasuper-Virasoro and fractional super-Virasoro algebras. These are two different generalizations of the ordinary super-Virasoro algebra generated by the infinitesimal diffeomorphisms of the superline. We work on the fractional superline parametrized by t and θ, with t a real coordinate and θ a paragrassmann variable of order M and canonical dimension 1/F . We further describe a more general structure labelled by M and F with M ≥ F . The case F = 2 corresponds to the parasuper-Virasoro algebra of order M , while the case F = M leads to the fractional super-Virasoro algebra of order F . The ordinary super-Virasoro algebra is recovered at F = M = 2. The connection with q-oscillator algebras is discussed.

Today, several applications require using electrostatic microphones in environments and/or in frequency ranges, which are significantly different from those they were designed for. When low uncertainties on the behavior of acoustic fields, generated or measured by these transducers, are required, the displacement field of the diaphragm of the transducers (which can be highly nonuniform in the highest frequency range) must be characterized with an appropriate accuracy. An analytical approach, which leads to results depending on the location of the holes in the backing electrode (i.e., depending on the azimuthal coordinate) not available until now (regarding the displacement field of the membrane in the highest frequency range, up to 100 kHz), is presented here. The holes and the slit surrounding the electrode are considered as localized sources described by their volume velocity in the propagation equation governing the pressure field in the air gap (not by nonuniform boundary conditions on the surface of the backing electrode as usual). Experimental results, obtained from measurements of the displacement field of the membrane using a laser scanning vibrometer, are presented and compared to the theoretical results.