Composite and janus type metallo-dielectric nanoparticles are increasingly considered as a means to control the spatial and temporal behavior of electromagnetic fields in diverse applications such as coupling to quantum emitters, achieve invisibility cloaks, and obtain quantum correlations between qubits. We investigate the surface modes of a toroidal nano-structure and obtain the canonical plasmon dispersion relations and resonance modes for arbitrarily layered nanorings. Unlike particle plasmon eigenmodes in other geometries, the amplitudes of the eigenmodes of tori exhibit a distinct forward and backward coupling. We present the plasmon dispersion relations for several relevant toroidal configurations in the quasistatic limit and obtain the dominant retarded dispersion relations of a single ring for comparison, discuss mode complementarity and hybridization, and introduce two new types of toroidal particles in the form of janus nanorings. The resonance frequencies for the first few dominant modes of a ring composed of plasmon supporting materials such as gold, silver, and aluminum are provided and compared to those for a silicon ring. A generalized Green's function is obtained for multilayer tori allowing for calculation of the scattering response to interacting fields. Employing the Green's function, the scalar electric potential distribution corresponding to individual poloidal and toroidal modes in response to an arbitrarily polarized external field and the field of electrons is obtained. The results are applied to obtain the local density of states and decay rate of a dipole near the center of the torus.

The small amplitude thermal vibrations of the microcantilever of an atomic force microscope can be enhanced via a delayed feedback system. This is verified experimentally for a triangular cantilever, and modeled theoretically as a boundary value problem resulting in a second order functional differential equation for the temporal behavior of the cantilever. The eigenvalues of the resulting delay differential equation describing the transverse vibrations of the cantilever are calculated and analyzed. These values are compared with the corresponding resonant frequencies predicted by a point mass model and with the experimentally observed values.

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A proof of an orthogonality relation for the MacDonald's functions with identical arguments but unequal complex lower indices is presented. The orthogonality is derived first via a heuristic approach based on the Mehler-Fock integral transform of the MacDonald's functions, and then proved rigorously using a polynomial approximation procedure.

The unbounded Bergman operator, the operator of multiplication by z on an unbounded open subset of the plane, is considered. We give a complete answer regarding the density problem of unbounded Bergman operators in terms of its equivalence to the problem of bounded point evaluations for the Bergman spaces. Using this equivalence and the notion of Wiener capacity, we obtain simple geometric conditions that classify almost those open subsets of the plane for which the corresponding Bergman operators are densely defined. With the aid of an analytic approach, we are also able to give condition for a large collection of open subsets of the plane for which all the positive integer powers of the corresponding Bergman operators are densely defined. Mathematics Subject Classification (2000). Primary 32A36; Secondary 47B38.Keywords. Unbounded Bergman operators, unbounded subnormal operators, density problem, bounded point evaluations, logarithmic and Wiener capacities.