Recursive interpolation,extrapolation and projection

The recursive projection algorithm derived in a previous paper is related to several well-known methods of numerical analysis such as the conjugate gradient method, Rosen's method and Henrici's. It is connected with the general interpolation problem, with extrapolation methods, with orthogonal projection on a subspace and with Fourier expansions. Several other connections and applications are presented.     

Prediction properties of some extrapolation methods

Convergence acceleration methods consist in the construction of a sequence converging faster than the initial sequence. Each member of the new sequence is a guess for the limit and it is computed from a restricted number of terms of the initial sequence. It is shown herein how convergence acceleration methods can be used to predict the next (unknown) term of the initial sequence instead of its limit. Particular emphasis on Aitken's △² process and the E-algorithm is placed.     

EPR Studies of Tl0.5Pb0.5Sr1.6Ba0.4CaCu2?xRuxO7?δ Superconductor

Bulk superconducting samples of type Tl_0.5Pb_0.5Sr_1.6Ba_0.4CaCu_2−x Ru _x O_7−δ, (Tl, Pb)/Sr-1212, with 0.0 ≤ x ≤ 0.525 were prepared by the conventional one-step solid-state reaction technique. The prepared samples were investigated using X-ray powder diffraction, electrical resistivity and electron paramagnetic resonance (EPR) measurements. Enhancement of the phase formation, superconducting transition temperature T _c and hole carriers concentration P was observed up to x = 0.075. For x > 0.075, a reverse trend was observed. EPR spectra were measured at different temperatures (120–290 K) for all prepared samples. The number of spins N participating in the resonance and the paramagnetic susceptibility χ were calculated as a function of both Ru-content and temperature. N and χ increased as the Ru-content increased. A linear relationship between logN and 1/T was established, from which the activation energy E _a was calculated as a function of the Ru-content. The temperature dependence of χ was fitted according to Curie–Weiss type of magnetic behavior. Curie constant C, Curie temperature θ, the effective magnetic moment μ and the electronic specific heat γ were estimated as a function of the Ru-content.     

Transpose-free Lanczos-type algorithms for nonsymmetric linear systems

The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB. This revised version was published online in June 2006 with corrections to the Cover Date.     

Limiting relationships and comparison theorems for sequences

The aim of this paper is to define some mathematical concepts which are useful to measure the speed of convergence of a sequence and to compare two converging sequences. In that way we define the order, the relative order and the α-equivalence of sequences. The asymptotic expansion of a series is studied and an application to Aitken acceleration process is given. A theorem similar to l'Hospital's rule is also proved for sequences.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Effect of IR laser radiation on the structure and optical properties of PC-PBT/Pd polymer nanocomposite films

ABSTRACT

Bayfol (PC-PBT blend ?lm) is a class of polymeric solid-state nuclear track detector which has a lot of applications in several radiation detection ?elds. It is a bisphenol-A polycarbonate PC blended with polybutylene terephthalate PBT. Bayfol/Palladium (PC-PBT/Pd) nanocomposite films have been deposited using the molding technique. It is worth mentioning that this report is almost the first one dealing with the topic of the changes of physical properties of Bayfol/Pd nanocomposite due to laser exposure. Samples from PC-PBT/Pd (5?wt%) nanocomposite were exposed to IR-pulsed laser of 5-W power, capable of producing 2000 pulses per second with pulse duration of 200?ns at 904?nm. The laser fluences were in the range 2–25?J/cm². The resultant modi?cations in the exposed nanocomposite samples have been studied as a function of fluence using different characterization techniques such as X-ray diffraction (XRD), UV spectroscopy and color difference studies. The results indicate the proper dispersion of Pd nanoparticles in the PC-PBT matrix that causes a strong intermolecular interaction between Pd and PC-PBT, resulted in an increase in refractive index and the amorphous phase. Also, it is found that the laser exposure reduces the optical energy gap that could be attributed to the increase in structural disorder of the exposed PC-PBT/Pd nanocomposites due to crosslinking. Further, the color intensity ΔE, which is the color difference between the exposed samples and the non-exposed one, was increased with increasing the laser fluence, convoyed by a significant increase in the green and yellow color components.