The Lyapunov dimension and its estimation via the Leonov method

Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of this method is that it allows one to estimate the Lyapunov dimension of invariant set without local- ization of the set in the phase space and in many cases get effectively exact Lyapunov dimension formula. In this survey the invariance of Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. An analog of Leonov method for discrete time dynamical systems is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works in the area: by Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (upper bounds of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (upper bounds of Hausdorff dimension via the Lyapunov exponents and dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.Comment: arXiv admin note: substantial text overlap with arXiv:1510.03835, Physics Letters A, 201

Neutrino dispersion in external magnetic field and plasma

Neutrino dispersion properties in an active medium consisting of magnetic field and plasma are analysed. We consider in detail the contribution of a magnetic field into the neutrino self-energy operator Sigma (p). The results for this contribution were contradictory in the previous literature. For the conditions of the early universe where the background medium consists of a charge-symmetric plasma, the pure magnetic field contribution to the neutrino dispersion relation is proportional to (e B)^2 and thus comparable to the contribution of the magnetized plasma. We consider one more hypothetical effect of the active medium influence on neutrino properties, the so-called ``neutrino spin light'' discussed in the literature. We show that this effect has no physical region of realization because of the medium influence on photon dispersion.Comment: 8 pages, LaTeX, no figures, based on the talk presented by A.V. Kuznetsov at the XIV International Seminar Quarks'2006, St.-Petersburg, Repino, Russia, May 19-25, 2006, to appear in the Proceeding

Compton-like interaction of massive neutrinos with virtual photons

The amplitude of a Compton-like process nu_i gamma^* -> nu_j gamma^* with virtual photons is calculated in the standard GWS theory with lepton mixing. The contribution of this process to the high energy neutrino scattering on the nucleus with single photon radiation nu N -> nu N gamma is discussed. The bremsspectrum and the total cross-section are estimated in the leading log approximation.Comment: 6 pages, LATEX, 2 PS figure

Could Vector Leptoquarks be Rather Light?

A possible Standard Model extension of the Pati-Salam type with a lepton number as the fourth color is reexamined. A new type of mixing in the interaction of the $SU(4)_V$--leptoquark with quarks and leptons is shown to be required. An additional arbitrariness of the mixing parameters could allow to decrease noticeably the lower bound on the leptoquark mass $M_X$ originated from the $\pi$ and $K$ decays and the $\mu e$ conversion. The only mixing independent bound emerging from the cosmological limit on the $\pi^0 \rightarrow \nu \bar{\nu}$ decay width is $M_X > 18~TeV$.Comment: (talk given at the VIII International Seminar "Quarks-94", Vladimir, Russia, May 11-18, 1994), 7 pages, LaTeX, uses world_sc.sty, Yaroslavl, Yaroslavl State University preprint YARU-HE-94/0

Neutrino chirality flip in a supernova and the bound on the neutrino magnetic moment

The neutrino chirality-flip process under the conditions of the supernova core is investigated in detail with the plasma polarization effects in the photon propagator taken into account. It is shown that the contribution of the proton fraction of plasma is essential. New upper bounds on the neutrino magnetic moment are obtained: mu_nu < (0.5 - 1.1) 10^{-12} mu_B from the limit on the supernova core luminosity for nu_R emission, and mu_nu < (0.4 - 0.6) 10^{-12} mu_B from the limit on the averaged time of the neutrino spin-flip. The best astrophysical upper bound on the neutrino magnetic moment is improved by the factor of 3 to 7.Comment: 7 pages, LaTeX, 5 EPS figures, based on the talk presented by N.V. Mikheev at the XIV International Seminar Quarks'2006, St.-Petersburg, Repino, Russia, May 19-25, 2006, to appear in the Proceeding

Comment on ``Spin light of neutrino in matter: a new type of electromagnetic radiation''

We show that a criticism made in hep-ph/0610294 against our paper Mod. Phys. Lett. A21, 1769 (2006) has no ground. We confirm that all results of investigations of the so-called ``spin light of neutrino (SL \nu)'' are incorrect because of the medium influence on photon dispersion. With taking account of this influence, both the SL \nu radiation rate and the total power are zero for neutrinos in all real astrophysical conditions.Comment: 5 pages, LaTeX, no figures, acknowledgements adde

Reply to Comment on ``Electron Mass Operator in a Strong Magnetic Field and Dynamical Chiral Symmetry Breaking''

This is our reply to comment (hep-ph/0206289) on our paper [Phys. Rev. Lett. 89 (2002) 011601].Comment: 1 page, REVTEX

A note on finite-time Lyapunov dimension of the Rossler attractor

For the R\"ossler system we verify Eden's conjecture on the maximum of local Lyapunov dimension. We compute numerically finite-time local Lyapunov dimensions on the R\"ossler attractor and embedded unstable periodic orbits. The UPO computation is done by Pyragas time-delay feedback control technique

Comment on the paper: ``Neutrino pair production by a virtual photon in an external magnetic field'' by Zhukovskii et al

We point out some serious mistakes in the investigation of Zhukovskii et al. Both the amplitude and the probability of the process were calculated wrongly, that is, the problem of the neutrino pair production by a virtual photon in an external magnetic field is still unsolved.Comment: 4 pages, LATE

Plasma induced neutrino spin flip via the neutrino magnetic moment

The neutrino spin flip radiative conversion processes nu_L -> nu_R + gamma^* and nu_L + gamma^* -> nu_R in medium are considered. It is shown in part that an analysis of the so-called spin light of neutrino without a complete taking account of both the neutrino and the photon dispersion in medium is physically inconsistent.Comment: 4 pages, LaTeX, 1 eps figure, based on the talk presented at the 13th Lomonosov Conference on Elementary Particle Physics, Moscow State University, Moscow, Russia, August 23-29, 2007, submitted to the Proceeding