Free subgroups of one-relator relative presentations

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G= always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in \~G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1. 4 page

Observation of large arrays of plasma filaments in air breakdown by 1.5-MW 110-GHz gyrotron pulses

We report the observation of two-dimensional plasma filamentary arrays with more than 100 elements generated during breakdown of air at atmospheric pressure by a focused Gaussian beam from a 1.5-MW, 110-GHz gyrotron operating in 3-mu s pulses. Each element is a plasma filament elongated in the electric field direction and regularly spaced about one-quarter wavelength apart in the plane perpendicular to the electric field. The development of the array is explained as a result of diffraction of the beam around the filaments, leading to the sequential generation of high intensity spots, at which new filaments are created, about a quarter wavelength upstream from each existing filament. Electromagnetic wave simulations corroborate this explanation and show very good correlation to the observed pattern of filaments.open424

de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article