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.     

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.     

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.     

Symmetrical triindole as a central unit in the fluorene-containing polyphenol assembly

A new strategy was proposed for the synthesis of fluorene-containing indole polyphenols, based on the Suzuki–Miyaura reaction of hexabrominated symmetric indole trimer with the monoboryl N,N'-di[9,9'-bis(3',5'-dimethoxybenzyl)fluoren-2'-yl]aniline derivative and subsequent chemical transformations. The prepared compound was applicable for the development of a promising positive resist for electron-beam nanolithography capable of forming patterns with a 12 nm resolution.     

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.     

Drying of colloidal solutions of fluoroalkyl oligomers

The change in the supramolecular structure upon drying (solvent removal) of colloidal solutions of fluoroalkyl oligomers at atmospheric pressure has been studied using atomic force microscopy. In an initial colloidal solution, micrometer-sized particles of the dense phase consist of randomly oriented oligomers in the form of rigid rods of a 3–5 nm length forming a porous framework filled with solvent molecules, which solvate the oligomer chains. The drying-induced capillary pressure, which in nanosized pores is of the same order of magnitude as the solvation energy, leads to framework deformation, collapse of the pores, and the formation of lamellar and dendritic structures on a 50–100 nm scale. The ordering of these structures (formation of blocks of parallel oriented fibers typical of a fluoroplastic) increases as the heat-treatment temperature and the drying rate are increased, increasing the roughness of the surface (ratio of real to smooth surface area) and its hydrophobicity.     

Visualization of supramolecular framework of perfluorinated-oligomer colloid particles by supercritical drying

Aerogels of the tetrafluoroethylene radical polymerization products H(C₂F₄)_nR, where R is the radical formed by the abstraction of a hydrogen atom from a solvent molecule, have been obtained by replacing the solvent with supercritical CO₂ and its subsequent rapid evaporation. According to the data of scanning electron and atomic force microscopy, the aerogel consists of loosely bound particles of 1–3 μm in diameter, which is two to three times that of colloid particles in the initial solution, where the particles consist of an oligomer framework filled with solvent molecules. The internal structure of the framework is manifested in the surface topography with a roughness coefficient of 1.6–1.8. High roughness leads to the formation of ultrahydrophobic coatings with contact angles of >160°. A model of supercritical drying in which the solvent is removed from the colloidal particles without alteration of the supramolecular structure is discussed.     

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.