Operatorial form of the theory of polarization optical devices: I. Spectral theory of the basic devices

While the theory of operators in quantum mechanics is expressed nowadays in a pure operatorial form (wrapped mostly in Dirac's symbolic language), in optics the polarization device operators and their action are analyzed yet in the old matrix (Jones or Muller) formalism. The theory of polarization device operators has not taken systematically advantage of the very general, fundamental and deep results of the spectral theory of operators, on the basis of which it can be structured in an elegant deductive and physically expressive form. In this paper we apply the spectral theorem to the polarization device operators, we calculate their expansions in a pure operatorial Dirac-dyadic form and give some examples which illustrate the advantages from the physical insight viewpoint of such an approach. We are concerning here only with the basic polarization devices, to which correspond normal operators.     

Canonical maps between semidirect products with applications to elasticity and superfluids

It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.     

On the generators of semisimple lie algebras

It will be shown that given any element X in a simple Lie algebra $\text{Q}$ over C, there exists a Y ∈ $\text{Q}$ such that the Lie algebra generated by X and Y is $\text{Q}$. The result is extended to the real semisimple Lie algebras. In some sense the main theorem of this paper can be regarded as an extension of Morozov-Jacobson theorem concerning three dimensional simple Lie algebras (see the remark at the end of Sec. 4). A new property of a special class of regular elements, known as the cyclic elements, is given.     

Selective transfer of energy through an alamethicin-doped artificial lipid membrane studied at discrete molecular level

In this study we present novel evidence that strengthens the paradigm of selective transfer of energy mediated by a random gating of ion channels. Specifically, we investigated the spectral response of a noisy artificial biomembrane whose electrical properties were largely dictated by embedded alamethicin oligomers. In this respect, we first evaluated experimentally the linear transfer function of the system via the white-noise analysis method. We prove that such a system displays specific ranges of frequency over which input signals pass preferentially, depending on their spectral content and the holding potential across the artificial bilayer which contains alamethicin. By employing voltage-driven periodic stimulation of alamethicin oligomers, we demonstrate that overall response of the system obeys qualitatively the predictions inferred from the transfer function analysis of it. These results emphasize the exquisite ability of excitable membranes to behave as band-limited filters and allow for maximal transfer of energy from an external stimulus over well-defined frequency ranges.