Some asymptotically optimal one-sided embeddings of trees of similar formulas into rectangular lattices

We consider the problem of optimally placing trees of formulas in rectangular lattices. We construct and study two types of these trees and corresponding ways of placing (embedding) them into such lattices. The first is based on perfect binary trees, while the second is based on special binary trees. For the second type of tree embeddings, we prove asymptotic optimality among the trees of all formulas similar to the initial formula of no greater depth.     

Fractional capacities relative to bounded open Lipschitz sets complemented

As a complement of (Shi and Xiao in Potential Anal 45:261–298, 2016) this article addresses five more aspects of the fractional capacities relative to bounded open Lipschitz subsets of \({\mathbb {R}}^n\): (i) quasi-everywhere version; (ii) unique extremal; (iii) energy functional; (iv) dual representation; (v) fine behaviour of each fractional Sobolev function in \({\mathbb {R}}^n\).     

Application of numerical schemes with singling out the boundary layer for the computation of turbulent flows using eddy-resolving approaches on unstructured grids

The use of eddy-resolving approaches to solving problems on arbitrary unstructured grids is investigated. The applications of such approaches requires the use of low dissipation numerical schemes, which can lead to numerical oscillations of the solution on unstructured grids. Numerical oscillations typically occur in domains with large gradients of velocities, in particular, in the near-wall layer. It is proposed to single out the boundary layer and use a numerical scheme with increased numerical dissipation in it. The algorithm for singling out the boundary layer uses a switching function to change the parameters of the numerical scheme. This algorithm is formulated based on the BCD scheme from the family NVD. Its validity and advantages are investigated using the zonal RANS–LES approach for solving some problems of turbulent flow of incompressible fluids.     

Approximation algorithm for the problem of partitioning a sequence into clusters

We consider the problem of partitioning a finite sequence of Euclidean points into a given number of clusters (subsequences) using the criterion of the minimal sum (over all clusters) of intercluster sums of squared distances from the elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is at the origin, while the center of each of the other clusters is unknown and determined as the mean value over all elements in this cluster. Additionally, the partition obeys two structural constraints on the indices of sequence elements contained in the clusters with unknown centers: (1) the concatenation of the indices of elements in these clusters is an increasing sequence, and (2) the difference between an index and the preceding one is bounded above and below by prescribed constants. It is shown that this problem is strongly NP-hard. A 2-approximation algorithm is constructed that is polynomial-time for a fixed number of clusters.     

Analytic continuation of the Appell function <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript> and integration of the associated system of equations in the logarithmic case

The Appell function F ₁ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F ₁ are related in a special way. Formulas for the analytic continuation of F ₁ beyond the unit bicircle are constructed in which F ₁ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F ₁ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.     

The effect of phosphate group replacement by sulfate groups on the phase formation in the synthesis of hydroxyapatite

Sulfate-substituted hydroxyapatite materials with a degree of substitution of up to 20 mol % (Ca₁₀(PO₄)_{(6 – 0.06x)}(SO₄)_0.09x (OH)₂, x = 0, 0.1, 0.5, 1, 5, 10, and 20) were synthesized. For substitutions of 0, 0.1, 0.5, and 1 mol %, a single-phase material with the apatite structure is formed. On further increase in the concentration of SO₄ ^2? groups up to 20 mol %, a second phase, CaSO₄, is formed; the amount of this phase increases for higher degrees of substitution. The unit cell parameters of hydroxyapatite-based materials change slightly upon the replacement of phosphate groups by sulfate groups: the parameter a tends to increase, while c tends to decrease. The introduction of sulfate groups results in decreasing particle size.     

New polymer composites based on keratin and polyethylene

New biodegradable composites based on keratin and polyethylene have been produced under shear deformation. It has been demonstrated that the introduction of keratin leads to an increase in elastic modulus and to a decrease in ultimate tensile strength and elongation at break of the compositions. Elongation at break ε_b depends on the keratin dispersity; the highest ε_b values are observed for the compositions containing the smallest keratin particles. It has been shown that the compositions are susceptible to mold fungi; i.e., they are biodegradable.     

Changes in luminescence of semiconductor colloidal quantum dots CdSe@CdS by replacement of hydrophobic ligands with 1-thioglycerol

Luminescent properties of CdSe@CdS colloidal quantum dots upon replacement of hydrophobic stabilizing ligands with water-soluble 1-thioglycerol have been studied. The data obtained have made it possible to optimize the ligand replacement procedure and significantly decrease the loss of the fluorescence quantum yield of the resulting hydrophilic quantum dots.     

Experimental study of gasification of vacuum residue in metal melt by alternating reactor feeding with carbonaceous feedstock and oxidant

Processes occurring in a metal melt during plasma-assisted melt gasification of carbonaceous feedstock have been studied in the mode of alternating feeding the reactor with the carbonaceous feedstock and oxidant. Vacuum residue was used as an oxygen-free carbonaceous material, and air and steam were used as an oxidant. The results of the study show that by alternating the carbonaceous-feedstock pyrolysis step and the step of carbon oxidation in the metal melt, it is possible to produce synthesis gas with any desired H₂/CO ratio.     

Conversion of hydrocarbon gases in dielectric barrier discharge in the presence of water

The conversion of C₁–C₄ hydrocarbons into gaseous and liquid products in a dielectric barrier discharge plasma in the presence of water has been studied. The formation of a deposit on the electrode surface is prevented by introducing water in the liquid state into a gaseous hydrocarbon stream, a finding that has been confirmed by IR spectroscopic study of the electrode surface. Hydrogen and C₂₊ hydrocarbons have been detected among the gaseous products of conversion, the liquid products being represented by C₆–C₁₀₊ alkanes. The total liquid products have amounted to 13.4, 26.0, or 36.6% for the methane, propane, or n-butane conversion, respectively. A 10% propane or butane admixture to methane increases the yield of the liquid products to make 22.0 and 31.7% for the methane–propane and the methane–butane mixture, respectively.