The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Dependence of Doubly Charged Ion Formation by the Two-Electron Mechanism on the intensity of Laser Radiation

Experimental dependences of the doubly charged ion yield N ²⁺ on the laser radiation intensity F are analyzed for the case of ion formation by the two-electron mechanism. It is shown that the region where the dependence N ²⁺(F) is characterized by the degree of nonlinearity K≈2.5 corresponds to the saturation of the process of doubly charged ion formation. This region is described well by theoretical calculations performed for Gaussian laser beams.     

Effective Values of Variable Components of the Refractive Index upon Nonlinear Recording of Phase Holograms

The diffraction of a plane wave incident on a thick hologram at the first and second Bragg angles in the process of nonlinear recording is analyzed using the coupled-wave equations. It is shown that, in this case, the two-wave approximation with the use of effective modulation amplitudes of the refractive index, n _1ef and n _2ef, can be applied. Analytical dependences of n _1ef and n _2ef on the modulation amplitudes of the refractive index n ₁ and n ₂ at the fundamental and doubled spatial frequencies are obtained and the criteria of their applicability are determined. The dependences found are used to calculate the parameters of gratings recorded in a photopolymeric material.     

Terahertz spectroscopy with a Josephson oscillator and a SINIS bolometer

The voltage response of a thin-film normal-metal hot-electron bolometer based on a SINIS (superconductor-insulator-normal metal-insulator-superconductor) structure to the radiation of a high-temperature Josephson junction in the terahertz frequency region was measured. Bolometers were integrated with planar log-periodic and double-dipole antennas, and Josephson junctions were integrated with log-periodic antennas. Measurements showed that the Josephson junction at a temperature of 260 mK was overheated by the transport current, so that its electron temperature exceeded 3 K at a bias voltage of 1 mV. The maximum response of a bolometer with a double-dipole antenna was observed at a frequency of 300 GHz, which agreed well with the calculated value. The Josephson radiation was observed at frequencies up to 1.7 THz. The voltage response of a bolometer reached 4×10⁸ V/W, and the total noise-equivalent power reached 1.5×10^?17 W/Hz^1/2.     

Excitation of low-lying nuclear states by the ion line emission in femtosecond laser plasma

A scheme of nuclear excitation by the ionic X-ray lines in laser plasma using two femtosecond laser pulses is proposed. The first pulse produces plasma with a given degree of ionization, allowing the X-ray line energies of the target ions to be tuned to resonance with the nuclear transition, while the second pulse generates hot electrons that are necessary for X-ray generation.     

Chern numbers for two families of noncommutative Hopf fibrations

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podle? spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

A solution to the problem of nonlinear surface vibration of a charged ideal liquid drop is found in a third-order approximation in initial multimode deformation of the equilibrium spherical shape by the method of many scales. It is shown that the spectrum of modes that are responsible for the shape of the drop at an arbitrary time instant depends considerably on the spectrum of modes governing the initial deformation of the drop. The latter spectrum also has an effect on nonlinear corrections to the vibration frequencies and, consequently, on a nonlinear correction to the critical Rayleigh parameter, which specifies the stability of the drop against self-charge.     

Transition radiation from relativistic charged particles interacting with atomic strings in a crystal

The problem of transition radiation generated by relativistic particles incident on atomic strings in a crystal at a small angle is considered. Conditions are obtained under which the problem of transition radiation reduces to that of radiation generated by a collision with a filament-like target. It is shown that the angular distribution of transition radiation is symmetric with respect to the atomic-string axis.     

Dynamic stability of a spiral domain in an ac magnetic field

The dependence of the stability of a magnetic spiral domain in a film on the parameters of the film and its domain structure and on an external magnetic field is considered within a phenomenological model. The model allows one to explain a number of experimentally observed properties of dynamic spiral domains resulting from the process of self-organization of domains and domain walls in an iron-garnet film placed in an external ac magnetic field.