Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Ukrainian Mathematical Journal - We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the...     

Integral Newton-Type Polynomials with Continual Nodes

We construct an integral Newton-type interpolation polynomial with a continual set of nodes. This interpolant is unique and preserves an operator polynomial of the corresponding degree.     

Stereoselective association of binuclear metallacycles in coordination polymers

A series of structurally related binuclear metallacycles [Cd(NO(3))(2)L](2), where L is an angular exo-bidentate ligand, have been synthesized. Each metallacycle contains two coordinatively unsaturated, chiral metal centers within a single molecule, and the assembly of these metallacycles into polymeric framework structures has been studied systematically for the first time. Stereoselective homochiral association of [Cd(NO(3))(2)L](2) leads to the formation of helical coordination polymers, whereas meso type association results in nonhelical chain structures. The type of stereoselective aggregation depends on the conditions of self-assembly as well as on ligand functionality. Both helical and nonhelical polymeric complexes have been isolated for the metallacycle [Cd(NO(3))(2)(2,4'-pyacph)](2) (2,4'-pyacph = 2,4'-(4-ethynylphenyl)bipyridyl). Homochiral association results in the formation of helical [Cd(NO(3))]( infinity ) chains which link the binuclear [Cd(NO(3))(2)(2,4'-pyacph)](2) metallacycles into racemic two-dimensional sheets which contain both P and M [Cd(NO(3))]( infinity ) helices. In contrast, meso-association leads to the formation of nonhelical one-dimensional chains. It is shown that the product of homochiral association is predominately formed at room temperature and that of meso-association is generated at elevated temperatures. Thus, it may be concluded that the homochiral association appears to be energetically less favorable than the meso-association, a conclusion that has been confirmed by theoretical calculations of the crystal lattice energy. Several high-yield syntheses of bipyridyl-type ligands used for metallacyclic assembly are also reported.     

Inorganic-organic interpenetrating frameworks: 4,4'-bipyridine N,N'-dioxide as a bridging hydrogen-bond acceptor

4,4'-Bipyridine N,N'-dioxide (L) acts as a hydrogen-bond acceptor in the compounds ([M(NO3)2(H2O)4].L2) (M = Co, Ni) to form doubly-interpenetrated framework materials with sixfold topological connectivity.     

Transition Metal Complexes of a Salen–Fullerene Diad: Redox and Catalytically Active Nanostructures for Delivery of Metals in Nanotubes

A covalently‐linked salen–C₆₀ (H₂L) assembly binds a range of transition metal cations in close proximity to the fullerene cage to give complexes [M(L)] (M=Mn, Co, Ni, Cu, Zn, Pd), [MCl(L)] (M=Cr, Fe) and [V(O)L]. Attaching salen covalently to the C₆₀ cage only marginally slows down metal binding at the salen functionality compared to metal binding to free salen. Coordination of metal cations to salen–C₆₀ introduces to these fullerene derivatives strong absorption bands across the visible spectrum from 400 to 630 nm, the optical features of which are controlled by the nature of the transition metal. The redox properties of the metal–salen–C₆₀ complexes are determined both by the fullerene and by the nature of the transition metal, enabling the generation of a wide range of fullerene‐containing charged species, some of which possess two or more unpaired electrons. The presence of the fullerene cage enhances the affinity of these complexes for carbon nanostructures, such as single‐, double‐ and multiwalled carbon nanotubes and graphitised carbon nanofibres, without detrimental effects on the catalytic activity of the metal centre, as demonstrated in styrene oxidation catalysed by [Cu(L)]. This approach shows promise for applications of salen–C₆₀ complexes in heterogeneous catalysis.     

Functionalized fullerenes in self-assembled monolayers

Anisotropy of intermolecular and molecule-substrate interactions holds the key to controlling the arrangement of fullerenes into 2D self-assembled monolayers (SAMs). The chemical reactivity of fullerenes allows functionalization of the carbon cages with sulfur-containing groups, thiols and thioethers, which facilitates the reliable adsorption of these molecules on gold substrates. A series of structurally related molecules, eight of which are new fullerene compounds, allows systematic investigation of the structural and functional parameters defining the geometry of fullerene SAMs. Scanning tunnelling microscopy (STM) measurements reveal that the chemical nature of the anchoring group appears to be crucial for the long-range order in fullerenes: the assembly of thiol-functionalized fullerenes is governed by strong molecule-surface interactions, which prohibit formation of ordered molecular arrays, while thioether-functionalized fullerenes, which have a weaker interaction with the surface than the thiols, form a variety of ordered 2D molecular arrays owing to noncovalent intermolecular interactions. A linear row of fullerene molecules is a recurring structural feature of the ordered SAMs, but the relative alignment and the spacing between the fullerene rows is strongly dependent on the size and shape of the spacer group linking the fullerene cage and the anchoring group. Careful control of the chemical functionality on the carbon cages enables positioning of fullerenes into at least four different packing arrangements, none of which have been observed before. Our new strategy for the controlled arrangement of fullerenes on surfaces at the molecular level will advance the development of practical applications for these nanomaterials.     

On interpolation approximation of differentiable operators in a Hilbert space

In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 554–563, April, 2006.     

Comparison of the stability of multiwalled carbon nanotube dispersions in water

Continuous real-time monitoring of the nanotube concentration in aqueous solution using UV-Vis spectroscopy allows quantitative comparison of the stability of different types of nanotube dispersions. Systematic investigation of the effects of nanotube length and functionalisation for thin multiwalled carbon nanotubes (MWNT) has revealed that shorter MWNT form more stable dispersions than longer nanotubes of the same diameter. MWNT shortened to an average length of approximately 1 microm form stable dispersions in water with concentrations up to 0.013 mg ml(-1) in the absence of surfactants or solubilising functional groups. The introduction of carboxylic or thiol groups on the surface of shortened nanotubes further increases the stability of MWNT dispersions (up to 0.24 mg ml(-1)). The introduction of surfactant or surface charge on MWNT has contrasting effects on functionalised and non-functionalised nanotubes, destabilising and stabilising their dispersions, respectively.