Orbital Instability of Standing Waves for the Klein-Gordon-Zakharov System

O Introduction We consider the orbital instability of standing waves for the Klein-Gordon-Zakharov system with different propagation speeds in three space dimensions     

On the basic transcendental functions appearing in the dielectric tensor of a weakly relativistic Maxwellian plasma

Summary It is shown how to deduce in a straightforward way the class of transcendental functions which appear in the elements of the dielectric tensor of a weakly relativistic thermal plasma. The deduction is made directly from the second form of Trubnikov integrals and is valid for arbitrarily large values of the wave perpendicular refractive index.

---

Riassunto Si mostra come sia possibile dedurre in modo molto diretto la famiglia di trascendenti che appare negli elementi della matrice dielettrica di un plasma debolmente relativistico in condizioni di equilibrio termico. Il procedimento parte direttamente dalla così detta seconda forma degli integrali di Trubnikov ed è valido per valori arbitrariamente grandi dell’indice di rifrazione perpendicolare al campo magnetico nel quale è immerso il plasma.

---

Резюме Мы показываем, как вывести непосредственным образом класс трансцендентных функций, которые появляются в элементах диэлектрического тензора для слабо релятивистской плазмы в условиях теплового равновесия. Вывод проводится непосредственно из второй формы интегралов Трубникова и справедлив для произвольно больших величин показателя преломления для волны, перпендикулярной магнитному полю, в которое помещена плазма.

     

Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures

In this paper, we explain how to associate a nonlinear martingale problem with some nonlinear parabolic evolution equations starting at bounded signed measures. Our approach generalizes the classical link made when the initial condition is a probability measure. It consists in giving to each sample-path a signed weight which depends on the initial position. After dealing with the classical McKean-Vlasov equation as an introductory example, we are interested in a viscous scalar conservation law. We prove uniqueness for the corresponding nonlinear martingale problem and then obtain existence thanks to a propagation of chaos result for a system of weakly interacting diffusion processes. Last, we study the behavior of the associated fluctuations and present numerical results which confirm the theoretical rate of convergence.     

Generalized loopy 2U: A new algorithm for approximate inference in credal networks

Credal networks generalize Bayesian networks by relaxing the requirement of precision of probabilities. Credal networks are considerably more expressive than Bayesian networks, but this makes belief updating NP-hard even on polytrees. We develop a new efficient algorithm for approximate belief updating in credal networks. The algorithm is based on an important representation result we prove for general credal networks: that any credal network can be equivalently reformulated as a credal network with binary variables; moreover, the transformation, which is considerably more complex than in the Bayesian case, can be implemented in polynomial time. The equivalent binary credal network is then updated by L2U, a loopy approximate algorithm for binary credal networks. Overall, we generalize L2U to non-binary credal networks, obtaining a scalable algorithm for the general case, which is approximate only because of its loopy nature. The accuracy of the inferences with respect to other state-of-the-art algorithms is evaluated by extensive numerical tests.     

Achieving Brouwer’s law with implicit Runge–Kutta methods

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times. AMS subject classification (2000)  65L06, 65G50, 65P10     

Propagation of a vectorial Laguerre–Gaussian beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of a vectorial Laguerre–Gaussian beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of the vectorial Laguerre–Gaussian beam is depicted in the reference plane. The light intensity distribution of a vectorial Laguerre–Gaussian beam with cos m is also compared with that of a vectorial Laguerre–Gaussian beam with sin m.     

Velocity of propagation in diffusional quantum theory

An equation of diffusional quantum theory which takes into account the finite velocity of propagation is derived from Kelvin's telegraph equation and Fürth's relation. The equation is then used to derive the ground state of quantum systems and to derive the Sommerfeld-Dirac expression for the ionization potential of hydrogen-like ions.     

Femtosecond second-order solitons in optical fiber transmission

A propagation of the femtosecond second-order solitons in an optical fiber is studied. We show that a generalized nonlinear Schrödinger equation well describes the propagation of the second-order soliton even containing only a few optical cycles. The propagations of a 50 fs and a 10 fs second-order soliton in an optical fiber are numerically simulated. It is found that, for the case of 10 fs second-order soliton, the soliton decay is dominated by the third-order dispersion, in contrast to the case of 50 fs second-order solitons, where the soliton decay is dominated by the delayed Raman response. It is also found that the exact delayed Raman response form must be used for the propagation of the 50 fs or less than 50 fs second-order soliton.