The best approximation of the Laplacian by bounded operators in the space L2

Рассматривается задача о наилучшем приближении оператора Лапласа первого порядка линейными ограниченными операторами с нормой, не превосходящей заданного числа, в пространстве L2(Rn) на классе функций, норма второй степени оператора Лапласа которых ограничена. Также в ходе решения этой задачи получена точная оценка нормы оператора Лапласа первого порядка через норму оператора Лапласа второго порядка и норму функции в пространстве L2(Rn).We consider the problem of the best approximation of the first order Laplace operator by linear bounded operators with norm not exceeding a given number in the space L2(Rn) on the class of functions with a bounded norm of the second degree of the Laplace operator. We also obtain an exact estimate for the norm of the first order Laplace operator in terms of the norm of the second order Laplace operator and the norm of the function in the space L2(Rn)

X-ray Diffraction Study of the NaF – LiF – LnF3 (Ln=La,Nd) Eutectic Composition in the Liquid and Solid States

The ternary eutectic alloy with the composition 33 mol % NaF+53 mol % LiF+14 mol % NdF3 has been studied in the liquid and solid states using differential thermal analysis and X-ray diffraction at room and high temperatures. The obtained results demonstrate that the eutectic alloy undergoes no phase transformations at temperatures from 298 K to melting point. Analysis of RMC models of the melt indicates that there is no dense non-crystalline packing of the fluorine anions in contrast to the eutectic NaF—LiF—LaF3 melt

Resonances and O-curves in Hamiltonian systems

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.Comment: 12 page

Investigation of the structure of CaO-Al2O3-SiO2 melts as a basis for the development of new agglomerated welding fluxes and industrial refractories

A significant number of welding fluxes and industrial ceramics are produced in the Ukraine. The production of these materials is sufficient for both domestic needs and significant export. It is assumed that the ternary CaO-Al2O3-SiO2 system may become the basis for the development of agglomerated welding fluxes and technical ceramics. Three samples of ternary CaO-Al2O3-SiO2 system were studied by means of high-temperature X-ray diffraction above the melting point: 23.3CaO- 14.7Al2O3-62.0SiO2 wt. % (sample 1, eutectic), 9.8CaO-19.8Al2O3-70.4SiO2 wt. % (sample 2, eutectic), 15.6 CaO- 36.5Al2O3-47.9SiO2 wt. % (sample 3). Experimental scattering intensity curves, structure factors and the radial distribution function of atoms were obtained. The structural parameters of short-range order were calculated using melt structure models obtained by Reverse Monte Carlo method. The existence of microparticle associates of the mullite or sillimanite type immersed in the slag matrix is assumed. It is assumed that the calcium ions coordinated aroundthese associates are surrounded by oxygen melt anions. Such a structure can be considered as a colloidal solution. The microparticle of mullite is the core of the aggregate that is surrounded by calcium and oxygen ions from the molten medium