Domains of attraction to Tweedie distributions

We derive new Tauberian theorems for natural exponential families, characterizing regularity properties of the family’s variance function in terms of those of an extreme generating measure. This provides normal and general domains of attraction to Tweedie distributions and leads to new results on weak convergence of natural exponential families to Tweedie distributions, parallel to weak convergence toward stable laws. In particular, we give the domains of attraction to the gamma and compound Poisson-gamma distributions.     

On the explicit parametric description of waves in periodic media

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.     

On the norms of generalized translation operators generated by Dunkl type operators

An integral representation for the generalized translation operators generated by Dunkl type operators $$ \Lambda f(x) = f'(x) + \frac{{A'(x)}}{{A(x)}}\frac{{f(x) - f\left( { - x} \right)}}{2} $$ In the spaces $ {L_p}\left( \mathbb{R} \right) $ with weight A is established, and a known estimate for their norms is improved. It is proved that under some natural assumptions on the functions A, these norms do not exceed two. Bibliography: 24 titles.     

Sequential single-shot imaging of nanoscale dynamic interactions with a table-top soft x-ray laser

We demonstrate the first real-space recording of nanoscale dynamic interactions using single-shot soft x-ray (SXR) full-field laser microscopy. A sequence of real-space flash images acquired with a table-top SXR laser was used to capture the motion of a rapidly oscillating magnetic nanoprobe. Changes of 30 nm in the oscillation amplitude were detected when the nanoprobe was made to interact with stray fields from a magnetic sample. The table-top visualization of nanoscale dynamics in real space can significantly contribute to the understanding of nanoscale processes and can accelerate the development of new nanodevices.     

Sliced linear zone plates for hard X-ray radiation

Linear zone plates for hard (E > 10 keV) X rays are considered. The technology of the plates includes cutting magnetron-sputtered W₅Si₃/Si multilayer coatings. The results of numerical computation suggest that this pair of materials provides a high efficiency in the first and second diffraction orders. It is found experimentally that good multilayer structures consisting of several hundreds of layers can be synthesized on silicon and glass substrates. It is shown that they can be cut by ion etching the multilayer coating through a mask.     

Research technique of metamaterials with the controlled effective negative refractive index and their characteristics

A design of the two-beam hybrid microwave interferometer for measuring refractive indices of metamaterials is described. It is based on the branching of radiation using a waveguide directional coupler. Radiations of the interferometer waveguide arms are summed and the phase disbalance of the interferometer arms is counted on the waveguide measuring line. The frequency dispersion of the negative refractive indices of the metamaterial samples with a variable distance between its constituent plates is experimentally investigated. A possibility of controlling the refractive index of a metamaterial is demonstrated.     

Molecular Structures of Dimeric and Trimeric Dineopentylaluminum Hydride and Trimeric Diphenylaluminum Hydride

The synthesis of dineopentylaluminum hydride, H‐Al(CH₂CMe₃)₂, is known in the literature since 1988. We determined the crystal structure of this important starting material and found different ring sizes with dimeric versus trimeric formula units. The molecular shape depends on the polarity of the solvent used for recrystallization (n‐pentane or 1,2‐difluorobenzene). The synthesis of diphenylaluminum hydride, which is also long known in the literature, was optimized. It has a trimeric structure in the solid state with an Al₃H₃ heterocycle.