Problems on a Semiaxis for an Integro-Differential Equation with Quadratic Nonlinearity

Computational Mathematics and Mathematical Physics - A functional equation is considered in which a linear combination of a two-variable function and its time derivative is set equal to the double...     

Mathematical Simulation of the Distribution of the Electron Beam Current during Pulsed Heating of a Metal Target

Journal of Applied and Industrial Mathematics - We study a model of the current distribution during heating of the surface of a tungsten sample under pulsed exposure to an electron beam. The model...     

Optimal cubature formulas in a reflexive Banach space

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.     

Coordination-based gold nanoparticle layers

Gold nanoparticle (NP) mono- and multilayers were constructed on gold surfaces using coordination chemistry. Hydrophilic Au NPs (6.4 nm average core diameter), capped with a monolayer of 6-mercaptohexanol, were modified by partial substitution of bishydroxamic acid disulfide ligand molecules into their capping layer. A monolayer of the ligand-modified Au NPs was assembled via coordination with Zr4+ ions onto a semitransparent Au substrate (15 nm Au, evaporated on silanized glass and annealed) precoated with a self-assembled monolayer of the bishydroxamate disulfide ligand. Layer-by-layer construction of NP multilayers was achieved by alternate binding of Zr4+ ions and ligand-modified NPs onto the first NP layer. Characterization by atomic force microscopy (AFM), ellipsometry, wettability, transmission UV-vis spectroscopy, and cross-sectional transmission electron microscopy showed regular growth of NP layers, with a similar NP density in successive layers and gradually increased roughness. The use of coordination chemistry enables convenient step-by-step assembly of different ligand-possessing components to obtain elaborate structures. This is demonstrated by introducing nanometer-scale vertical spacing between a NP layer and the gold surface, using a coordination-based organic multilayer. Electrical characterization of the NP films was carried out using conductive AFM, emphasizing the barrier properties of the organic spacer multilayer. The results exhibit the potential of coordination self-assembly in achieving highly controlled composite nanostructures comprising molecules, NPs, and other ligand-derivatized components.     

Assembly of coordination nanostructures via ligand derivatization of oxide surfaces

A scheme is presented for the construction of coordination nanostructures on oxide surfaces (glass, Si/SiO2, quartz), based on application of epoxy-terminated monolayers as anchors for covalent grafting of ligands. Two ligands bearing amine groups were reacted with epoxysilane monolayers on oxide surfaces, providing ligand-terminated substrates. The ligands employed were (i) a pyridine moiety, used for subsequent binding of cobalt tetraphenylporphine (CoTPP), and (ii) deferoxamine (DFX), which contains hydroxamic acid moieties, used for subsequent construction of various Zr4+-based coordination layers. The results suggest that a dense ligand layer was obtained in both cases, allowing the formation of coordination overlayers on the oxide surfaces. The growth of coordinated layers was similar to analogous overlayers assembled on Au substrates, indicating that high ligand coverage is achieved by the epoxy-amine surface reaction. Epoxy-based functionalization of oxide substrates is a mild and efficient method for preparing high-quality coordination overlayers. Moreover, the method makes use of commercially available silane and amine reactants, providing the basis for wide application.     

Preparation of graded materials by laterally controlled template synthesis

A novel scheme is presented for the synthesis of graded materials by electrodeposition in porous insulating templates. Lateral control of copper electrodeposition in nanoporous alumina membranes is achieved by application of a lateral potential gradient on a thin Au film evaporated on the membrane, used as the cathode. Formation of metal gradients in the membranes is shown to occur under conditions where essentially no gradient is formed on similar bare electrodes. This is attributed to the permanent resistivity of the thin Au film between the pores, which does not disappear upon Cu deposition, allowing a potential gradient to be maintained. Formation of a copper gradient in porous alumina membranes by uniform deposition followed by gradient dissolution is also demonstrated. These results establish the feasibility of controlled electrodeposition and gradient formation in nanoporous insulating templates.     

The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces

We give an upper bound for the deviation of the norm of a perturbed error from the norm of the original error of a cubature formula in a multidimensional bounded domain. The deviation arises as a result of the joint influence on the computations of small variations of the weights of a cubature formula and rounding in the subsequent calculations of the cubature sum in the given standards (formats) of approximation to real numbers. We estimate the practical error of a cubature formula acting on an arbitrary function from the unit ball of a normed space of integrands. The resulting estimates are applied to studying the practical error of cubature formulas in the case of integrands in Sobolev spaces on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for cubature formulas constructed as the direct product of quadrature formulas of rectangles along the edges of the unit cube. The weights of this direct product are positive.