Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure U_q(sl(2)) \odot U_{\qhat}(sl(2)) is a non trivial extension of $U_q(sl(2))$ and U_{\qhat}(sl(2)). We construct the corresponding generalized exponentials and the generalized Liouville field.Comment: 36 pages, LaTex, LPTENS 93/4

A Model of Polarisation Rotations in Blazars from Kink Instabilities in Relativistic Jets

This paper presents a simple model of polarisation rotation in optically thin relativistic jets of blazars. The model is based on the development of helical (kink) mode of current-driven instability. A possible explanation is suggested for the observational connection between polarisation rotations and optical/gamma-ray flares in blazars, if the current-driven modes are triggered by secular increases of the total jet power. The importance of intrinsic depolarisation in limiting the amplitude of coherent polarisation rotations is demonstrated. The polarisation rotation amplitude is thus very sensitive to the viewing angle, which appears to be inconsistent with the observational estimates of viewing angles in blazars showing polarisation rotations. Overall, there are serious obstacles to explaining large-amplitude polarisation rotations in blazars in terms of current-driven kink modes.Comment: 6 pages, 3 figures; Proceedings of the conference "Polarised Emission from Astrophysical Jets", 12-16 June 2017, Ierapetra, Greece; Eds. M. Boettcher, E. Angelakis and J. L. G\'{o}me

The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an old version of uuencode that produces blank characters some times suppressed by the mailer. Same content

Minimum time control of a nonlinear system

Time-optimal control problem studied for system representing second-order nonlinear differential equatio

Skew-product maps with base having closed set of periodic points

In [Proc. ECIT-89, World Scientific, (1991), 177–183], A. N. Sharkovski˘ı and S.F. Kolyada stated the problem of characterization skew-product maps having zero topological entropy. It is known that, even under some additional assumptions, this aim has not been reached. In [J. Math. Anal. Appl., 287, (2003), 516–521], J. L. G. Guirao and J. Chudziak partially solved the problem in the class of skew-product maps with base map having closed set of periodic points. The present paper has two aims for this class of maps, on one hand to improve that solution showing the equivalence between the property “to have zero topological entropy” and the fact “not to be Li-Yorke chaotic in the union of the ω-limit sets of recurrent points”. On other hand, we show that the properties “to have closed set of periodic points” and “all nonwandering points are periodic” are not mutually equivalent properties, for doing this we disprove a result from Efremova of 1990