Characteristic Operator of a Diffusion Process

Semi-Markov processes of diffusion type in the d-dimensional space (d > 1) are considered. We assume that the transition generating function of such a process satisfies a second-order differential equation of elliptic type. We apply methods of differential equations theory, especially of the theory of the Dirichlet problem, to study the transition generating function for a small neighborhood of the initial point of the process. Asymptotic expansions in the small scale parameter are obtained both for the first exit point distribution density and for the first exit time expectation for the case where the trajectory of the process leaves a small neighborhood of the initial point. We prove the existence of the Dynkin characteristic operator determined by a decreasing sequence of neighborhoods. Bibliography: 9 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 293, 2003, pp. 226–251.     

Variational Derivative for Buckling Loads of Beams and Frames

The finite-difference method is a numerical technique for obtaining approximate solutions to differential equations. The main objective of the present study is to give a new aspect to the finite-difference method by using a variational derivative. By applying this formulation, accurate values of the buckling loads of beams and frames with various end supports are obtained. The performance of this formulation is verified by comparison with numerical examples in the literature __________ Published in Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 139–144, July 2005.     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × _R (where _R is the real axis), we construct a monotone decreasing stochastic field η_ρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field η_ρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 92–143.     

Electronic structure of the diaquabis(hydrogen malato) copper(II) crystal

Summary The electronic absorption and e.s.r. spectra of a crystal of the title compound [Cu(H₂O)₂ (C₄H₅O₅)₂] were recorded and experimental results are discussed quantitatively, using ligand field theory and the radial wave function of bound Cu^II. The electronic structure of the compound is consistent with its crystal structure.     

Solid-state Electrochemistry of Gadolinium Hexacyanoferrate Modified Electrode and Electrocatalytic Properties of Gadolinium Hexacyanoferrate

A new electroactive polynuclear inorganic compound of rare earth metal, gadolinium hexacyanoferrate (GdHCF), was prepared and characterized using the techniques of FTIR spectroscopy, thermogravimetric analysis (TG), UV-Vis spectrometry, X-ray photoelectron spectroscopy (XPS), ICP atomic emission spectroscopy, and EDX. The results of ICP atomic emission spectroscopy, EDX, and TGA indicated that the prepared GdHCF sample had a stoichiometry of NaGdFe(CN)₆·12H₂O (when GdHCF was prepared in NaCl solution). The FTIR spectrum of GdHCF showed that there were two types of water molecules in the structure of GdHCF: one was the interstitial water (5 H₂O), which resulted from the association of water due to H-bonding, and the other was water coordinated with Gd (7 H₂O). The results obtained using XPS showed that the oxidation state of Fe and Gd in the GdHCF sample was +2 and +3, respectively. GdHCF was immobilized on the surface of spectroscopically pure graphite (SG) electrode forming the GdHCF/SG electrode, and the solid-state electrochemistry of the resultant electrode was studied using cyclic voltammetry. The cyclic voltammetric results indicated that the GdHCF/SG electrode exhibited a pair of well-defined and stable redox peaks with the formal potential of E^0′=(197±3) mV. The effects of the concentration of the supporting electrolyte on the electrochemical characteristics of GdHCF were studied, and the results showed that the value of E^0′ increased linearly with the activity of the cationic ion of the supporting electrolyte (lga_Na⁺), with a slope of 54.1 mV, which may become a novel method for determining the activity of Na⁺ in solution. Further experimental results indicated that GdHCF had electrocatalytic activities toward the oxidation of dopamine (DA) and ascorbic acid (AA), and the electrocatalytic current increased linearly with the concentration of DA (or AA) in the range of 1.0–10.0 mmol·L^?1 (for DA) or 0.5–20.0 mmol·L^?1 (for AA).     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.