Starlikeness and Certain Class of Rational Functions

A new criterion for starlikeness in the unit disc and an application to a certain class of rational functions are given.     

Nano-antenna-assisted harmonic generation

We report on low-order harmonic generation utilising the plasmonic field enhancement in arrays of rod-type gold optical antennae. Furthermore, we examine their suitability to support high-order harmonic generation (HHG). The low-order harmonics are used as a tool to investigate the nonlinear properties of the antennae. Particular attention is paid to the thermal properties, which become significant at the peak intensities necessary for HHG. A theoretical model explains the experimental findings and enables future improvements. In experiments we observe up to the fifth harmonic order and measure a field enhancement sufficient to support high-order harmonic generation. Moreover, we find a damage threshold for the antennae.     

Tamm oscillations in semi-infinite nonlinear waveguide arrays

We demonstrate the existence of nonlinear Tamm oscillations at the interface between a substrate and a one-dimensional waveguide array with either cubic or saturable, self-focusing or self-defocusing nonlinearity. Light is trapped in the vicinity of the array boundary due to the interplay between the repulsive edge potential and Bragg reflection inside the array. In the special case when this potential is linear these oscillations reduce themselves to surface Bloch oscillations.     

Observation of discrete gap solitons in one-dimensional waveguide arrays with alternating spacings and saturable defocusing nonlinearity

We consider, both experimentally and theoretically, the existence and stability of localized, symmetric, and antisymmetric gap solitons (GSs) in binary lattices of identical waveguides but with alternating spacings. Furthermore, the properties of surface GSs at the boundary of the lattice are explored.     

A synthesis of N-alkyl and N,N-dialkyl O-ethyl thiocarbamates from diethyl dixanthogenate using different oxidants

Abstract  

A novel synthesis of N-alkyl and N,N-dialkyl O-ethyl thiocarbamates from diethyl dixanthogenate and primary and secondary amines, using three oxidizing systems, has been developed on the laboratory scale, and the method using sodium hypochlorite has been applied on a semi-industrial scale. The effect of the oxidizing agents, sodium hypochlorite, in-situ-generated peracetic acid, and the manganese(II) acetate/oxygen system on product purity and yield was studied. The results obtained by use of these three methods were compared with those obtained by reaction of sodium ethyl xanthogenacetate and amines, and of sodium ethyl xanthate with amines in the presence of sulfated nickel zeolite catalyst. The reaction mechanism of sodium hypochlorite oxidation has been established on the basis of isolation of reaction intermediates and determination of their structure by use of Fourier-transform infrared, ¹H and ¹³C NMR, and mass spectrometric methods. The suggested sodium hypochlorite and manganese(II) acetate/oxygen systems have many advantages in comparison with commercial and catalytically promoted synthetic methods, because they are new ecologically friendly syntheses.     

On an inequality of Friedrich's type

In this paper the dependence of the constant in the inequality on simply connected bounded domains is found.

     

Estimates for the norm of powers of Volterra's operator through its singular values

We prove that for every Volterra operator T the inequality holds, where b is an absolute constant and are its singular values. Received January 26, 2000 / Published online February 5, 2001     

Dynamics of bright discrete staggered solitons in photovoltaic photorefractive media

Theoretical results on spatial optical bright solitons excited in arrays of nonlinear defocusing waveguides, that result from the photovoltaic effect in a photorefractive material, are presented. The existence of four types of stationary discrete bright staggered solitons, on-site, inter-site, twisted inter-site, and twisted on-site solitons, is shown both analytically and numerically, and their stability properties are investigated. The maximum Hamiltonian of staggered solitons with the same total power corresponds to stable modes. It is shown that for low total power the on-site mode is stable while in the high power regime the inter-site mode is stable. These results are confirmed numerically. In addition, steering properties of localized modes are investigated by introducing a transversal translational shift. Because of the translational symmetry between on-site and inter-site localized modes they are considered as two dynamical realizations of the same moving mode, and the formalism of the Peierls-Nabarro effective potential is applied to interpret the exchange between trapping and steering of these modes. This critically depends on the mode’s total power and the introduced phase difference. On the other hand, steering of twisted inter-site and on-site localized modes is not numerically observed. Instead, transversal perturbation leads to a transformation of twisted modes either into a trapped on-site mode of smaller power and radiation, or into two trapped on-site modes.     

Regularized trace of the inverse of the dirichlet laplacian

For the eigenvalues $( \lambda_{n}) _{n=1}^{\infty}$ of the Dirichlet Laplacian on a bounded convex domain $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\Omega\subset{\C}$ , we find the sum of the series the regularized trace of the inverse of Dirichlet Laplacian. © 2011 Wiley Periodicals, Inc.