Nanophotonics and nanometrology with planar X-ray waveguide-resonator

The mechanism of X-ray waveguide-resonance propagation or the radiation superstream model, which can become the ground of X-ray nanophotonics, is discussed briefly. Some attention is devoted to features consideration of the simplest devices characterized by the waveguide-resonance transportation of X-ray beams. The experimental data showing the user possibilities of a simplest waveguide-resonators application for diffractometry are presented. We discuss the main reasons to improve the metrological characteristics for total reflection X-ray fluorescence (TXRF) analytical method in case when the target exciting beam is formed by a waveguide-resonator. Some problems appearing during the waveguide-resonator application are formulated.     

Self-similar planar curves related to modified Korteweg-de Vries equation

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg-de Vries equation of a given (small) mean.     

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.     

A complete set of numerical invariants for a subfactor

We show that certain numerical invariants associated naturally to a subfactor planar algebra constitute a complete family in the sense of determining the isomorphism class of the subfactor planar algebra.In the course of the proof, we show also that planar algebra isomorphisms of subfactor planar algebras can always be chosen to be ∗-preserving. This latter statement generalises the fact that ‘Hopf algebra isomorphisms of finite-dimensional Kac algebras can be chosen to be ∗-preserving’.     

Design and synthesis of novel imidazolium-based ionic liquids with a pseudo crown-ether moiety: diastereomeric interaction of a racemic ionic liquid with enantiopure europium complexes

A planar-chiral imidazolium salt with a tris(oxoethylene) bridge was synthesized, and its potential application as a room temperature ionic liquid with a molecular-recognition ability was demonstrated.     

The design of molecules containing planar tetracoordinate carbon

A novel preference for planar tetracoordination was observed over the conventional tetrahedral arrangement in a new series of C₅H₂, C₅H₄, C₅H₄^1+/2+ and related compounds. The stability of these molecules is assessed with the ring-opening barriers, HOMO-LUMO gap, singlet-triplet energy differences and nucleus independent chemical shift values.     

Miniaturization of separation techniques using planar chip technology

Miniaturization of separation columns implies equally reduced vol- umes of injectors, detectors, and the connecting channels. Planar chip technology provides a powerful means for the fabrication of micron-sized structures such as channels. This is demonstrated by two examples. An optical absorbance detector chip exhibits the expected behavior of a 1 mm optical path length cell despite its volume of 1 nL. A capillary electrophoresis device allows integrated injections of 100 pL samples, efficiencies of 70,000 to 160,000 theoretical plates in 10 to 20 seconds, and external laser-induced fluorescence detection at any capillary length of choice between 5 and 50 mm.     

A compact formula for the coupling efficiency associated with mirrors in waveguide lasers

This paper presents a simple theory based on the diffraction integral approach for calculating the coupling efficiency in waveguide lasers which employ plane or spherical mirrors. The results obtained differ only slightly from a full rigorous approach, due to the fact that only less than one percent of the beam energy lies outside the waveguide aperture. The infinite aperture approximation employed here leads to an explicit compact formula for the coupling efficiency, whereas exact calculations lead to expressions requiring numerical integration for their evaluation.     

Is planarity of pyridin-2-yl- and pyrazin-2-yl-formamide thiosemicarbazones related to their tuberculostatic activity? X-ray structures of two pyrazine-2-carboxamide-N′-carbonothioyl-hydrazones

Crystal structures of two title compounds and several their relatives known earlier reveal conservative and characteristic features, which may be related to their tuberculostatic activity. The molecules are predominantly planar due to conjugation through five successive bonds in the zwitterionic fragment S⁻–C(sp²)–N–NH⁺–C(sp²)–NH₂ and intramolecular hydrogen bonds, which prevent rotation of the adjacent pyrazine (or pyridine) ring. It has been suggested that in spatial sense such planar molecules resemble acridines intercalating with nucleic acids and that similar process may be responsible for tuberculostatic activity of the title pyrazine-2-carboxamide-N′-carbonothioyl-hydrazones.