Schmidt analysis of pure-state entanglement

We examine the application of Schmidt mode analysis to pure-state entanglement. Several examples permitting exact analytic calculation of Schmidt eigenvalues and eigenfunctions are included, as well as evaluation of the associated degree of entanglement.     

On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory

We demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type. The corresponding variational principle is constructed using the theory of equilibrium of a potential in an external field. Dedicated to the memory of B. M. Levitan Supported by RFBR grants Nos. 05-01-00522 and NSh-1551.2003.1, by Program No. 1 of the Branch of Mathematics, Russian Academy of Sciences, and by INTAS project No. 03-51-6637.     

Observation of the nonlinear effect in relativistic Thomson scattering of electron and laser beams

Thomson scattering of high-power laser and electron beams is a good test of electrodynamics in the high-field region. We demonstrated production of high-intensity X-rays in the head-on collision of a CO₂ laser and 60-MeV electron beams at Brookhaven National Laboratory, Accelerator Test Facility. The energy of an X-ray photon was limited at 6.5 keV in the linear (lowest order) Thomson scattering, but the nonlinear (higher order) process produces higher energy X-rays. We measured the angular distribution of the high-energy X-rays and confirmed that it agrees with theoretical predictions.     

Advantages of NIR radiation use for optical determination of skin horny layer thickness with embedded TiO2 nanoparticles during tape stripping procedure

The tape stripping technique is an experimental method frequently used for reconstruction of the in-depth distribution of various topically administered substances within the horny layer of human skin, e.g., compounds contained in sunscreens. Titanium dioxide (TiO₂) nanoparticles (25–200 nm in diameter) are one such compound. Optical techniques which apply blue light are found to be suitable for reconstruction. However, the presence of particles affects the light propagation within the skin and therefore causes incorrect determination of strip thickness, leading to an improper reconstructed distribution of nanoparticle concentration revealed from the experimental data. This study evaluates the errors emerging from the use of blue (400 nm) and NIR (800 nm) radiation and finds the use of longer wavelength light more advantageous. Particles of different diameters are considered, and it is revealed that the application of small particles (25–60 nm) results in the lowest rate of error.     

Measurement of the Seperation between Atoms beyond Diffraction Limit

We propose a scheme to obtain the distance of two identical atoms placed inside the standing wave field by monitoring the collective resonance fluorescence spectrum emitted by the two particles. We find three different parameter ranges, depending on the distance of the atoms as compared to the transition wavelength. For large interparticle distances, dipole-dipole coupling is negligible, and the main system evolution arises from the interaction with the standing wave field. In the small-distance limit, the dynamics is dominated by the dipole-dipole interaction. Finally, in the intermediate region, a rich interplay of the various couplings arises, which however is lifted for strong driving laser fields. The present measurement procedure allows us to distinguish the three cases. In each of the cases, we show how to determine the distance of the two particles and their respective positions relative to the nodes of the standing wave field with fractional-wavelength precision.     

Direct Renormalizations of the One-Dimensional Torus

Renormalization formulas connecting the points of the orbits O ₀ ^± , i.e., the sequences generated by an irrational shift on the half-open interval [0, 1) or (0, 1], and the points of the orbits' derivatives d^mO ₀ ^± , i.e., the sequences obtained by restricting O ₀ ^± to half-open intervals of smaller length, are obtained. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 142–154.     

On construction of symmetries from integrals of hyperbolic partial differential systems

An algorithm is proposed which allows one to construct higher symmetries of arbitrary order for some special classes of hyperbolic systems possessing integrals. The Pohlmeyer-Lund-Regge system and the open two-dimensional Toda lattices are shown to belong to the class of systems where our algorithm is applicable. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.