On the Symmetry of Extremals in the Weight Embedding Theorem

Varying conditions on the weight function f, the author is interested in whether the extremal of the ratio $\inf \frac{{\left\| {\nabla u} \right\|_{Lp} \left( \Omega \right)}}{{\left\| {fu} \right\|_{Lp} \left( \Omega \right)}}{\lambda }_{pq} \left( {\Omega ,f} \right)$ remains symmetric. Here, f is positive almost everywhere in $u \in \mathop {W_p^1 }\limits^{\text{O}} \left( \Omega \right)$ and the infimum is taken over all functions $u \in \mathop {W_p^1 \left( \Omega \right)}\limits^{\text{O}}$ . Bibliography: 23 titles.     

Asymptotics of a Solution to the Quasiliniear Neumann Problem in a Neighborhood of a Conical Point

The existence of a small solution to the quasilinear Neumann problem is established in a H\"older class with separated asymptotics. The lower-order asymptotic terms are constructed. The homogeneity exponents of these terms turn out to be dependent on the value of the solution at a conical point. Bibliography: 33 titles.     

Experimental investigations of nonlinear acoustic phenomena in polycrystalline zinc

The results of experimental and theoretical investigations of nonlinear acoustic phenomena (nonlinear losses, shift of resonance frequency, generation of the third harmonic, and nonlinear sound-by-sound damping) in polycrystalline zinc nonannealed and annealed resonating rods are presented. The measurements were carried out in the 10(-7)-10(-5) strain range at a frequency of about 3 kHz; the frequency of the weak ultrasonic pulse was about 270 kHz. The experimentally observed phenomena are described in frames of phenomenological equations of state containing elastic hysteresis and dissipative nonlinearity. The nonlinear acoustic parameters of these equations are determined by comparison between theoretical dependencies and experimental results. The influence of structural changes in zinc due to annealing on the nonlinear acoustic phenomena is shown.     

ECSim2.0 package and simulation of X-ray emulsion chamber experiments

The ECSim 2.0 package is being developed on the basis of the ECSim software package to simulate the development of nuclear-electromagnetic cascades initiated by different particles in a layered medium. Some characteristics of ECSim 2.0 and the results obtained using this package in simulation of the passage of electrons and γ rays through the gamma block of X-ray emulsion chambers in the Pamir Collaboration experiment are discussed. The results obtained are compared with the data of other researchers.     

Investigation of heat transfer from a lead heat carrier to a tube streamlined longitudinally

Results of experimental studies on the local characteristics of heat transfer from a lead heat carrier to the surface of a cooled tube in an annular gap are shown at control and alteration of oxygen admixture content under the conditions of power circuits with heavy liquid-metal heat-transfer agent. This work is aimed at obtaining the grounded formulas for engineering calculations of heat transfer surfaces. Investigations were carried out at the lead temperature of 400–500 °C, the average velocity of heat-transfer agent of 0.1–1.5 m/s, the range of Prandtl number of 0.0123–0.0211 and Peclet numbers of 500–7000. The heat flux changed within 50–160 kW/m². Controllable changing content of oxygen admixture changed from the value of thermodynamic activity of oxygen from 10⁻⁵-10⁰ to saturation and higher with deposition of lead oxides near the heat-transferring surface.     

The effect of acoustic nonlinearity on nonlinear wave processes in solids: The inverse problem

The perturbation method is used to study nonlinear acoustic effects in solids that feature elastic, hysteresis, or inelastic nonlinearity. It is found that these media exhibit qualitatively different nonlinearity behavior laws, which facilitates classifying them based on the acoustic nonlinearity type.     

Sol–Gel Synthesis of Membrane Mo<Subscript>2</Subscript>C/Al<Subscript>2</Subscript>O<Subscript>3</Subscript> Catalysts with Different Architectures and Their Catalytic Activity in the Reaction of Carbon Dioxide Conversion of Methane

The methods of synthesis of composite membrane catalysts based on Mо₂C and Al₂O₃ support by a sol–gel method were developed. The samples of membrane catalysts with different architectures were developed. The difference in the catalytic activity of membrane catalysts in carbon dioxide conversion of methane depending on the porous structure and morphology of the catalytic layer was studied.     

The Asymptotic Properties of the Solution to the Stokes Problem in Domains That Are Layer-Like at Infinity

Asymptotic formulae are derived for solutions to the Stokes problem in domains which, outside a ball, coincide with the three-dimensional layer \Bbb R² ×(0,1) {\Bbb R}^2 \times (0,1) . The properties of detached asymptotic terms differ in the transversal and longitudinal directions. In order to justify the asymptotic expansions the procedure of dimension reduction is employed together with estimates for miscellaneous weighted norms of the solutions.     

Lp-Estimates for a Solution to the Dirichlet Problem and to the Neumann Problem for the Heat Equation in a Wedge with Edge of Arbitrary Codimension

The Dirichlet problem and the Neumann problem in a wedge with edge of an arbitrary codimension are studied. On the basis of the Green functions of these problems in a cone, estimates for solutions are obtained. Coercive estimates for the solutions are also obtained in the Kondrat'ev spaces. Bibliography: 14 titles.