Nonlocal vertex algebras generated by formal vertex operators

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundations for this study. For any vector space W, we study what we call quasi compatible subsets of Hom (W,W((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with W as a natural faithful quasi module in a certain sense, and that any quasi compatible subset generates a nonlocal vertex algebra with W as a quasi module. In particular, taking W to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with W as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.     

On a conjecture of Fefferman and Graham

In 1985, Fefferman and Graham have constructed a local embedding of an arbitrary real-analytic manifold, of odd dimension n, into a Ricci-flat manifold of dimension n+2 admitting a homothety. They conjectured that their result remains valid in even dimensions, if logarithms are allowed in the expansion of the metric. In this paper, we (i) prove that such expansions exist and converge and (ii) establish the degree of non-uniqueness of the solution in terms of the coefficients in the expansion.     

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

Long Abelian ideals

We study Abelian ideals of a Borel subalgebra consisting of long roots. It is shown that methods of Cellini and Papi can be extended to this situation. A uniform expression for the number of long Abelian ideals is given. We also show that there is a one-to-one correspondence between the long Abelian ideals and B-stable commutative subalgebras in the little adjoint representation of the Langlands dual Lie algebra.     

Normalization of complex-valued planar vector fields which degenerate along a real curve

Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.     

Rank estimation in reduced-rank regression

Reduced rank regression assumes that the coefficient matrix in a multivariate regression model is not of full rank. The unknown rank is traditionally estimated under the assumption of normal responses. We derive an asymptotic test for the rank that only requires the response vector have finite second moments. The test is extended to the nonconstant covariance case. Linear combinations of the components of the predictor vector that are estimated to be significant for modelling the responses are obtained.     

Sound velocity in alumino-silicate liquids determined up to 2550 K from Brillouin spectroscopy: glass transition and crossover temperatures

Hypersonic longitudinal sound velocities in five silicate and alumino-silicate liquids have been measured between 293 and 2550 K by Brillouin spectroscopy. Together with previous observations for four other glasses and liquids of the system SiO₂-Al₂O₃-CaO-MgO, these results are used to discuss changes in hypersonic velocities in three adjacent temperature domains, i.e., below, in, and above the glass transformation range. The temperature dependence of Brillouin velocities is consistent with the observed variations with temperature of viscosity, density, and mean heat capacity for the same three temperature domains. These variations of physical properties of alumino-silicate liquids are qualitatively in agreement with the Inherent Structure Theory for liquids.     

Model spectral function and superconductivity in BaxK1−xBiO3 (x=0.5-0.7)

The Eliashberg spectral density function is constructed for the superconducting compound Ba_xK_1−xBiO₃ with x=0.5, 0.6, 0.7. The functions, besides yielding some prominent structures, show widely spreading spectra that inhibit lattice instability by yielding not too large values of λ(<1.32). The resulting spectral functions show similar shape as other results, where available, and also successfully reproduce the observed critical temperature, isotope exponent, gap ratio and several other parameters.