Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions

A Global Optimization Algorithm for Multivariate Functions with Lipschitzian First Derivatives

In this paper we propose a new multi-dimensional methodto solve unconstrained global optimization problems with Lipschitzianfirst derivatives. The method is based on apartition scheme that subdivides the search domain into a set of hypercubesin the course of optimization. This partitioning is regulated by thedecision rule that provides evaluation of the "importance"of each generated hypercube and selection of some partition element to performthe next iteration. Sufficient conditions of global convergence for the newmethod are investigated. Results of numerical experiments are alsopresented.     

Boundary Properties of Second-Order Partial Derivatives of the Poisson Integral for a Half-Space ℝk+1 + (k>1)

The boundary properties of second-order partial derivatives of the Poisson integral are studied for a half-space ^k+1 ₊.     

Conformational studies of substituted nitroanilines: geometry of 2-methyl-5-nitroaniline

The title compound (C₇H₈N₂O₂), is monoclinic, space group P2₁/n, witha=9.552(2),b=5.677(2),c=13.586(3)Å, =92.68(2)⁰, andD _x=1.374 g-cm^–3 forZ=4. The refinement converged toR=0.043,wR=0.038. The molecule is approximately planar, with dihedral angles of 3.7(2.1)⁰ between the amino group and the aromatic ring, and 3.2(2)⁰ between the nitro group and the ring. According to the UV spectrum in solution, the molecular geometry indicates weak intramolecular charge transfer. The three-dimensional structure is stabilized by three intermolecular H bonds. A bifurcated one induces the formation of chains along , while the other two link molecules that belong to adjacent chains and are related by an inversion center.     

Synthesis and X-ray structural analysis of (2<Emphasis Type="Italic">Z</Emphasis>)-3(4-hydroxyphenyl)-2-pyridin-4-ylacrylonitrile and (2<Emphasis Type="Italic">Z</Emphasis>)-3-(3,5-di-<Emphasis Type="Italic">tert</Emphasis>-butyl-4-hydroxyphenyl)2-pyridin-4-ylacrylonitrile

Synthesis and X-ray structural investigations have been carried out for the two title compounds C₁₄H₁₀N₂O (3a) and C₂₂H₂₆N₂O (3b). Compound 3a crystallizes in the monoclinic space group P2₁/c, with a = 3.843(1) Å, b = 24.618(5) Å, c = 11.318(2) Å, = 92.61(3), V = 1069.7(4) Å,³ and Z = 4. Compound 3b crystallizes in the triclinic space group P-1, with a = 9.004(2) Å, b = 9.447(2) Å, c = 11.713(2) Å, = 76.70(3), = 83.12(3), = 82.16(3), V = 956.5(3) Å,³ and Z = 2. Both stilbazole derivatives have Z-geometry about the ethylene bridge which links the heterocyclic and aromatic rings. The molecular skeleton of 3a is slightly non-planar: the dihedral angles between the acrylonitrile linkage and the pyridine ring, and between this linkage and the p-hydroxyphenyl ring are 7.2(2) and 4.1(2), respectively. The molecular skeleton of 3b is less planar: the values of similar dihedral angles are 17.0(2) and 20.8(2), respectively. In the crystal of 3a, the molecules are packed in stacks along the a axis with head-to-head orientation. Intermolecular hydrogen bonds O=H s N and C=H s N link molecules into sheets parallel to (100) plane. In the crystals of 3b, the molecules have a head-to-tail orientation and intermolecular hydrogen bonds O=H s N link the molecules into infinite chains along [01-1] direction.     

Molecular structures of M(tfac)3 (M=Al, Co) and Cu(H2O)(fod)2: Examples of unusual supramolecular architecture

The molecular structures of Al(tfac)₃ (1), Co(tfac)₃ (2) (H-tfac = 1,1,1-trifluoroacetylacetone) and Cu(H₂O)(fod)₂ (3) (H-fod = 1,1,1,2,2,3,3-hepta-fluoro-7,7-dimethyloctane-4,6-dione) have been determined. The metal coordination spheres in compounds 1 and 2 are essentially the same as the respective M(acac)₃ derivatives. Despite the isomorphous nature of the structures of compounds 1 and 2, the identity of the nearest intermolecular van der Waals contacts are altered by minor changes in the metal coordination sphere. The geometry about copper in compound 3 is close to that of an ideal square bipyramid with the -diketonate ligands occupying the basal plane. The water ligand in each molecule of compound 3 is hydrogen bonded to an oxygen of a -diketonate ligand on an adjacent molecule resulting in the formation of dimers, which form rods along the y-axis due to weak C–F···Cu interactions. Crystal data: (1) orthorhombic, Pca2₁, a = 14.949(3), b = 19.806(4), c = 13.624(3) Å, V = 4033(1) ų, and Z = 8, and (2) orthorhombic, Pca2₁, a = 14.930(3), b = 19.620(4), c = 13.540(3) Å, V = 3966(1) ų, and Z = 8,; (3) monoclinic, P2₁/c, a = 12.447(3), b = 10.486(2) c = 21.980(4) Å, = 102.65(3)°, V = 2799(1) ų, and Z = 4.     

Conformations of 2-C:1-N-carbonyl-2-deoxy-d-glycopyranosylamines

Geometry and molecular conformation of theN-toluenesulfonyl1 andN-acetyl-2 derivatives of peracetylated 2-C:1-N-carbonyl-2-deoxy--d-glucopyranosylamine were investigated with the use of X-ray diffraction methods. Compound1 (C₂₀H₂₃NSO₁₀) crystallizes in the monoclinic P2₁ space group, withZ=2 anda=8.238(1),b=7.988(1),c=16.928(2)Å, =99.12(1)°. Compound2 (C₁₅H₁₉NO₉) crystallizes in orthorhombic P2₁2₁2₁ space group withZ=4 anda=8.385(1),b=8.550(1),c=24.000(2) Å. Analysis of differences in bond lengths and angles between compounds1 and2 and other compounds of this class showed that the electronwithdawing effect by the residue located at the nitrogen atom can be manifested by lengthening of the -lactam C-N bonds, with simultaneous shortening of the distance between two carbon atoms at the ring fusion. Semi-empirical calculations suggested that the title compounds displayed two positively charged centers, susceptible for attack of nucleophiles, one at the carbonyl group of -lactam and the second at the anomeric carbon atom. Atomic charges, however, calculated for compounds1 and2 did not explain their different reaction directions during alcoholysis.Part 3. For part 1 and 2, see Refs. 5 and 6.     

Microwave-assisted solvent free synthesis of hydroxy derivatives of 4-methyl coumarin using nano-crystalline sulfated-zirconia catalyst

Nano-crystalline sulfated-zirconia solid acid catalyst has been studied for microwave-assisted solvent free synthesis of hydroxy derivatives of 4-methyl coumarin by Pechmann reaction. The catalyst showed good activity for activated m-hydroxy phenol substrates, viz., phloroglucinol and pyrrogallol with ethyl acetoacetate for the synthesis of 5,7-dihydroxy 4-methyl coumarin and 7,8-dihydroxy 4-methyl coumarin, respectively, showing significant yields ranging from 78 to 85% within 5–20 min at 130 °C. However, the less activated phenol and m-methyl phenol was observed to be inactive for the synthesis of 4-methyl coumarin and 4,7-dimethyl coumarin, respectively, under the studied experimental conditions.     

Crystal structure and thermoanalytical study of a manganese(II) complex with 1-allylimidazole

The crystal structure of a manganese(II) 1-allylimidazole complex ([Mn(1-AIm)₃(NO₃)₂], where 1-Aim=1-allylimidazole), was characterized by X-ray diffraction (XRD) using SHELX-97. The thermal behaviour of the complex was investigated by thermogravimetry (TG) coupled with an FTIR unit. The complex showed a multi-step decomposition related to the release of the ligand molecules, followed by oxidation. The final residue at 1073 K was found to be manganese(II) oxide. Evolved gas analysis allowed to prove the oxidative decomposition pattern of the examined complex, initially proposed by the percentage mass loss data. Finally, a kinetic analysis of the oxidative decomposition steps was made using the Kissinger equation, while the complex nature of the decomposition kinetics was revealed by the isoconversional Ozawa-Flynn-Wall method.     

Fluorophenyl derivatives of elements II-VI groups

The enthalpies and temperatures of melting of RSi(CH₃)₃, R₄Si, R₃P, R₃As, R₃Sb, R₃Bi, R₂Te and R₂Hg (R=C₆F₅) were obtained by scanning calorimetry measurements. The pressure of the saturated and unsaturated vapors of RSi(CH₃)₃, R₂Si(CH₃)₂, R₄Si, R₃Ga, R₃P, R₃As, R₃Sb, R₃Bi, R₂Te and R₂Hg has been measured by the static method with a membrane-gauge manometer. It was established that all investigated substances proceeded to vapor as monomers. Equations approximating the dependences of saturated vapor pressures on temperature and the enthalpies and entropies of vaporization were obtained. Grafite films with silicon intercalated up to 25 at.% were grown by CVD using R₄Si as a precursor. These films showed semiconductor properties in the temperature interval 80–300 K.