Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Polycondensation Products of Phloroglucine with Difunctional Aliphatic Amines

The known reaction of condensation of a strong amine with phloroglucine under quite mild conditions was the starting point for synthesis of a new series of macromolecular resins not previously dèscribed. Phloroglucine has been condensed with a series of nine aliphatic diamines in aqueous solution under atmospheric pressure. The resins obtained are dark powders, which can be molded under increased pressure in the temperature range of 175–310°C. The thermal, mechanical, chemical, and electrical properties of the new resins have been examined and their structure confirmed. Some properties of resins can be calculated by means of approximate mathematical formulas.     

Effective Conductivity of Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants

The aim of this paper is a study of the quasilinear transport equation, for instance the stationary heat equation. For periodically microheterogeneous media asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Golden–Papanicolaou integral representation theorem and some bounds developed for the linear equation have been extended. Two-point Padé approximants have been used to calculate bounds. Examples are also provided.     

Estimation of a Stieltjes function expanded to Taylor series at complex conjugate points

Taylor series expansions of a Stieltjes function f in various complex conjugate points are used to construct the so called unified continued fractions (UCF) terminated on P-th step by a remainder named tail of f. We prove that, if f is a Stieltjes function then its tail is also a Stieltjes function. The estimations of f are obtained in what follows. Numerical calculations of the new complex bounds on f generated by complex conjugate input data are carried out.     

Formation and Thermal Decomposition of Indium Oxynitride Compounds

During the reactions of lithium oxide with indium nitride, lithium nitride with indium oxide, and lithium nitride with lithium indate LiInO2, the formation of a previously unknown crystalline phase, of composition Li4InNO2, was observed. The course of thermal decomposition of the new compound was determined. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multipoint matrix Padé approximant bounds on effective anisotropic transport coefficients of two-phase media

It is well known that a tensor Stieltjes function f represents an effective transport coefficient q of an inhomogeneous medium consisting of two isotropic components. In this paper, we investigate multipoint matrix Padé approximants to matrix expansions of f. We prove that matrix Padé ones to f estimate f from the top and below. Consequently the Padé approximants to q form upper and lower bounds on q. The inequalities for matrix Padé bounds on f and q are established. They reduce to the inequalities for scalar Padé ones Tokarzewski (ZAMP 61:773–780, 2010). As an illustrative example, matrix Padé estimates of an effective conductivity of a specially laminated two-phase medium are computed.     

Two-point Padé approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Padé approximants to Stieltjes function has been extended to the case of certain two-point Padé approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.