Sur le théorème de Lévy–Raikov–Marcinkiewicz

Cataloged from PDF version of article.Let μ be a finite non-negative Borel measure. The classical Lévy-Raikov-Marcinkiewicz theorem states that if its Fourier transform μ̂ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then μ̂ admits analytic continuation into the strip {t: 0<It<R}. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. © 2004 Elsevier Inc. All rights reserved

On the Levy-Raikov-Marcinkiewicz theorem

Cataloged from PDF version of article.Let μ be a finite non-negative Borel measure. The classical Lévy-Raikov-Marcinkiewicz theorem states that if its Fourier transform μ̂ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then μ̂ admits analytic continuation into the strip {t: 0<It<R}. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. © 2004 Elsevier Inc. All rights reserved

Direct reconstruction of the two-dimensional pair distribution function in systems with angular correlations

An x-ray scattering approach to determine the two-dimensional (2D) pair distribution function (PDF) in partially ordered 2D systems is proposed. We derive relations between the structure factor and PDF that enable quantitative studies of positional and bond-orientational (BO) order in real space. We apply this approach in the x-ray study of a liquid crystal (LC) film undergoing the smectic-hexatic phase transition, to analyze the interplay between the positional and BO order during the temperature evolution of the LC film. We analyze the positional correlation length in different directions in real space.Comment: 23 pages, 8 figure

Modulational Instability and Complex Dynamics of Confined Matter-Wave Solitons

We study the formation of bright solitons in a Bose-Einstein condensate of $^7$Li atoms induced by a sudden change in the sign of the scattering length from positive to negative, as reported in a recent experiment (Nature {\bf 417}, 150 (2002)). The numerical simulations are performed by using the 3D Gross-Pitaevskii equation (GPE) with a dissipative three-body term. We show that a number of bright solitons is produced and this can be interpreted in terms of the modulational instability of the time-dependent macroscopic wave function of the Bose condensate. In particular, we derive a simple formula for the number of solitons that is in good agreement with the numerical results of 3D GPE. By investigating the long time evolution of the soliton train solving the 1D GPE with three-body dissipation we find that adjacent solitons repel each other due to their phase difference. In addition, we find that during the motion of the soliton train in an axial harmonic potential the number of solitonic peaks changes in time and the density of individual peaks shows an intermittent behavior. Such a complex dynamics explains the ``missing solitons'' frequently found in the experiment.Comment: to be published in Phys. Rev. Let

Modulational instability in nonlocal Kerr-type media with random parameters

Modulational instability of continuous waves in nonlocal focusing and defocusing Kerr media with stochastically varying diffraction (dispersion) and nonlinearity coefficients is studied both analytically and numerically. It is shown that nonlocality with the sign-definite Fourier images of the medium response functions suppresses considerably the growth rate peak and bandwidth of instability caused by stochasticity. Contrary, nonlocality can enhance modulational instability growth for a response function with negative-sign bands.Comment: 6 pages, 12 figures, revTeX, to appear in Phys. Rev.