Homogenization of Some Problems with Rapidly Oscillating Constraints

Homogenization theorems are proved for the problem of minimization of a quadratic integral functional on a set of admissible functions satisfying rapidly oscillating periodic constraints on the boundary of a perforated domain. At each point of the perforation surface, the trace of any admissible function belongs to a given segment of the real axis, and this segment periodically varies along the boundary. According to the structure of this field of segments on the boundary, we characterize the asymptotic behavior of the solutions (minimizers) as the period of the structure tends to zero.     

Killing tensors as irreducible representations of the general linear group

We show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorphic to a certain highest weight, irreducible representation of the general linear group. The isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup. As an application, we deduce the Delong–Takeuchi–Thompson formula on the dimension of the vector space of Killing tensors from the classical Weyl dimension formula. To cite this article: R.G. McLenaghan et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Stokes eigenmodes in square domain and the stream function–vorticity correlation

The Stokes eigenmodes in the square are numerically determined and their symmetry properties are identified. The spectra evolution laws are in excellent qualitative agreement with the theoretical asymptotic predictions proposed by Constantin and Foias (in “Navier–Stokes equations”, University of Chicago Press, 1988), . The slopes are reported here and are found to be specific to the eigenmodes symmetry family. The dynamic equilibria are analyzed and show a linear relationship between the vorticity and the stream function in the core of the eigenmodes. These features of the Stokes eigenmodes confined in the square are shared by the fully periodic Stokes eigenmodes.     

Some Further Properties of the Accelerated Kerr-Schild Metrics

We extend the previously found accelerated Kerr-Schild metrics for Einstein-Maxwell-null dust and Einstein-Born-Infeld-null dust equations to the cases including the cosmological constant. This way we obtain the generalization of the charged de Sitter metrics in static space-times. We also give a generalization of the zero acceleration limit of our previous Einstein-Maxwell and Einstein-Born-Infeld solutions.     

Plasmon resonance in a nanosphere

The phenomenon of plasmon resonance in a nanosphere and a nanospheroid is considered. As is known, this phenomenon does not depend explicitly on the size of these nanoparticles. However, it is shown that, actually, the resonance conditions are determined by the diameter of the nanosphere and by the length of the major axis of the nanospheroid.     

Recording of dynamic holograms by nanosecond and picosecond laser pulses in solid-state fullerene-containing matrices

The results of a complex investigation of the recording of dynamic holograms by pulsed radiation with a duration of 20 ns and 300 ps in fullerene-containing media based on porous glasses and polymethyl methacrylate matrices are reported. Spectral analysis of these media in the range 300–700 nm was performed. The efficiency of hologram recording was found to relate with the occurrence of a band at 330 nm in the absorption spectrum of a medium. It is shown that dynamic holograms can be recorded in the fullerene-containing media by nanosecond and picosecond pulses. In the latter case, holograms arise mainly due to a change in the electronic polarizability of fullerene molecules, which is almost inertialess. The results obtained can be used in the development of ultrafast switching devices based on dynamic holograms.     

Mathematical modeling of the Magnus effect in the geometrical optics approximation

The Magnus effect in multimode fibers with triangular and parabolic refractive index profiles (RIPs) is simulated in the geometrical optics approximation. The calculations confirm the linear relation between the angle of rotation Δ? and the fiber length z. The results of calculations for a spiral path with constant radius are compared with the analytical solution obtained. For a fiber with a parabolic RIP, the value of Δ? obtained in this work is one-half the result obtained in the wave approach.     

Electronic excitation energy transfer between dye molecules in H<Subscript>2</Subscript>O and D<Subscript>2</Subscript>O solutions in the micellar phase

Processes of electronic excitation energy transfer (EEET) in H₂O+D₂O solutions within reversed micelles with different degrees of hydration were studied. It is ascertained that the structure of solutions forms a cluster distribution of interacting molecules having a fractal dimensionality. As a result, in regions with a locally high concentration of the dye molecules, an increase in the EEET efficiency is observed. Solubilization by reversed micelles leads to destruction of the cluster structure of the water system, with the dependence of the EEET efficiency on the degree of hydration being nonmonotonic. As the degree of hydration increases, the inhomogeneities of the structure inherent in bulk water are restored.