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.     

The relationship between self-organization and membrane effects of aqueous dispersion systems of the thyroliberin oligopeptide

The relationship between rearrangement of the dispersed phase inducing considerable changes in the pH and nonmonotonic concentration dependences of membrane effects in aqueous systems of the endogenous regulatory peptide, thyroliberin (thyrotropin-releasing hormone), in 10^–3–10^–16 mol/L concentration range was demonstrated for the first time. The membrane structure modification in the 10^–13–10^–16 mol/L range was found to be due to accumulation of nanoassociates, while the oppositely directed pronounced structural changes in the 10^–6–10^–12 mol/L range may be associated with the coexistence and rearrangement of dispersed phases of various nature (domains and associates) whose action on membrane lipid components is regulated in this concentration range by the correlated changes in the dispersed phase parameters and pH.     

Electronic properties of achiral and chiral gold nanotubes

The band structure of ten single-walled gold nanotubes of different radius and chirality angle have been calculated by the linearized augmented cylindrical wave method. For all tubes, the Fermi level crosses the half-filled band; therefore, the tubes are characterized by a metallic electronic structure. The band structure of the nanotubes changes relatively weakly with a change in nanotube structure. The valence band width for all the tubes is 9.1 eV. The density of states at the Fermi level remains unaltered with a change in chirality angle and decreases by 30% with an increase in radius from 3 to 12 Å.     

Effect of 3<Emphasis Type="Italic">d</Emphasis>-metal dopants on the electronic properties of hexagonal titanium dioxide nanotubes

The electronic structure of hexagonal TiO₂ nanotubes doped with 3d transition metals from Sc to Zn was calculated by the linearized augmented cylindrical wave method. The calculated densities of states demonstrate that the substitution of Sc, V, Co, Cu, or Fe atoms for a part of Ti atoms leads to the decrease in the band gap width of the material from 4 to 2 eV. Such nanotubes are promising materials for creation of electrodes for electrochemical photolysis of water.     

Phase equilibria in the Cu–Cu<Subscript>2</Subscript>Se–As system

Phase equilibria in the Cu–Cu₂Se–As were investigated by differential thermal analysis and X-ray powder diffraction analysis. Informative plots describing this system were constructed, viz., the polythermal sections Cu_0.667Se_0.333–As, Cu_0.667Se_0.333–Cu_0.735As_0.265, and Cu_0.8Se_0.2–As, the isothermal section of the phase diagram at 300 K, and the projection of the liquidus surface. The obtained results differ from the published data in length of fields of primary crystallization of phases and in coordinates of a number of invariant equilibrium points.     

Copper(II), nickel(II), and zinc(II) complexes with <Emphasis Type="Italic">o</Emphasis>-tozylaminobenzaldehyde 4,6-dimethylpyrimidyl hydrazone

o-Tosylaminobenzaldehyde dimethylpyrimidyl hydrazone and its copper(II), nickel(II), and zinc(II) complexesare synthesized and studied. According to the X-ray diffraction data, the zinc(II) complex has the structure of a tetragonal pyramid with the N₃O donor ligand environment. According to EPR and magnetochemistry data, the copper(II) complex has a similar structure of the chelate core. In the nickel(II) complex, the solvent molecule completes the coordination core to an octahedron.     

Study of lithium complexes with benzo-15-crown-5 ether by X-ray diffraction and IR spectroscopy

The complex formation of lithium with benzo-15-crown-5 (B15C5) was investigated. The complexes LiB15C5H₂OX, where X = Cl^? (1), I^? (2), (3), (5), and LiBF₄B15C5 (4) were synthesized and studied by IR spectroscopy. Complexes 1–4 were examined by X-ray diffraction. According to IR spectroscopy data, the crown ether conformation changes upon dissolution. The interaction of the extracted complex with the solvent was identified.     

Crystal structure of EuLnAgS<Subscript>3</Subscript> (Ln = Gd and Ho) compounds

The crystal structures of the first prepared EuLnAgS₃ (Ln = Gd and Ho) compounds, which have two polymorphs, were determined by X-ray powder diffraction. α-EuLnAgS₃ phases are isostructural to BaErAgS₃ (monoclinic crystal system, space group C2/m): a = 17.3168(10) Å, b = 3.9683(2) Å, c = 8.3174(4) Å, β = 103.94° (EuGdCuS₃); a = 17.1729(12) Å, b = 3.9367(3) Å, c = 8.2905(6) Å, β = 103.9° (EuHoCuS₃). β-EuLnAgS₃ phases belong to the AgBiS₂ structure type (cubic crystal system, space group Fm-3m): a = 5.739(2) Å (EuGdCuS₃) and a = 5.678 Å (EuHoCuS₃). In the α-EuLnAgS₃ crystal structure, LnS₆ octahedra and AgS₅ trigonal bipyramids share edges to form a three-dimensional (3D) structure with channels accommodating Eu²⁺ ions. A decrease in Ln³⁺ ionic radius gives rise to the crystal-chemical contraction of the 3D structure.     

Two crystalline polymorphic forms of α-(<Emphasis Type="Italic">N</Emphasis>-benzoxazolin-2-one)acetic acid

Two crystalline polymorphic forms of α-(N-benzoxazolin-2-one)acetic acid (BAA) are prepared by changing the temperature of its crystallization from solution in ethanol. Crystallographic data of the α-form are determined: a = 12.7769(17) Å, b = 8.2574(9) Å, c = 16.7390(19) Å, β = 105.087(13)°, space group C2/c, V = 1705.2(4) ų, and Z = 8, while those of β form are a = 5.2854(4) Å, b = 5.9880(4) Å, c = 13.4509(5) Å, β = 94.666(4)°, space group P2₁, V = 424.30(4) ų, and Z = 2. It is found that BAA molecules of the α form combine into infinite one-dimensional chains arranged along axis b by means of O?H···O and C?H···O hydrogen bonds, and these chains are crosslinked via C?H···O hydrogen bonds to form a threedimensional structure. The β form has another system of hydrogen bonds, one of which is bifurcated (O4···O2, O4···O3), and the π–π-interactions between the benzoxazolinone fragments of BAA molecules combined into a chain also arranged along axis b are observed. Calorimetric analysis shows that the polymorphic transition from the α form to the β form occurs at 129°C.