Finite Element Matrix Sequences: the Case of Rectangular Domains

In the present paper, we consider a preconditioning strategy for Finite Element (FE) matrix sequences {A _n (a)} _n discretizing the elliptic problem $$\left\{ \begin{gathered} A_a u \equiv ( - )^k \nabla ^k [a(x,y)\nabla ^k u(x,y)] = f(x,y),{ }(x,y) \in \Omega = (0,1)^2 , \hfill \\ \left. {\left( {\frac{{\partial ^s }}{{\partial v^s }}u(x,y)} \right)} \right|_{\partial \Omega } \equiv 0,{ }s = 0,...,k - 1,{ }^{^{^{^{^{^{(1)} } } } } } \hfill \\ \end{gathered} \right.$$ with a(x,y) being a uniformly positive function and ν denoting the unit outward normal direction. More precisely, in connection with preconditioned conjugate gradient (PCG) like methods, we define the preconditioning sequence: {P _n (a)} _n , P _n (a):= $$\widetilde D$$ _n ^1/2(a)A _n (1) $$\widetilde D$$ _n ^1/2(a), where $$\widetilde D$$ _n (a) is the suitable scaled main diagonal of A _n (a). In fact, under the mild assumption of Lebesgue integrability of a(x), the weak clustering at the unity of the corresponding preconditioned sequence is proved. Moreover, if a(x,y) is regular enough and if a uniform triangulation is considered, then the preconditioned sequence shows a strong clustering at the unity so that the sequence {P _n (a)} _n turns out to be a superlinear preconditioning sequence for {A _n (a)} _n . The computational interest is due to the fact that the computation with A _n (a) is reduced to computations involving diagonals and two-level Toeplitz structures {A _n (1)} _n with banded pattern. Some numerical experimentations confirm the efficiency of the discussed proposal.     

Cold rolling mill process: a numerical procedure for industrial applications

The paper proposes a model for a cold rolling mill process in the full-film regime that uses lubricant emulsion sprayed on at the entrance of the strip. The aim of the model is to forecast the reduction of strip thickness versus the flow rate of lubricant given the other operation parameters. The model includes strip plastic deformation, lubricant flow and lubricant viscosity depending on pressure. The mathematical problem is a free boundary one and a numerical procedure, applied to an industrial plant, is presented with some results.     

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations: Application to Orthogonal Polynomials

This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite–Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite–Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.     

A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes

A reliable Monte Carlo method for the evaluation of first passage times of diffusion processes through boundaries is proposed. A nested algorithm that simulates the first passage time of a suitable tied-down process is introduced to account for undetected crossings that may occur inside each discretization interval of the stochastic differential equation associated to the diffusion. A detailed analysis of the performances of the algorithm is then carried on both via analytical proofs and by means of some numerical examples. The advantages of the new method with respect to a previously proposed numerical-simulative method for the evaluation of first passage times are discussed. Analytical results on the distribution of tied-down diffusion processes are proved in order to provide a theoretical justification of the Monte Carlo method.     

An improved scheme for accelerating the convergence of the simulated annealing algorithm applied to real world problems

When formulated in mathematical terms, the problem of zoning a protected natural area subject to both box and spatial constraints results in a large combinatorial optimization problem belonging to the NP-hard class. These facts suggest the need to apply a heuristic approach. In this contribution a new proposal to decrease the control parameter, known as temperature, in the simulated annealing algorithm is presented. The strategy is based on that proposed by Lundy and Mees [4], and developed in order to decrease the running time of the algorithm applied to large scale problems. When applied to solving small-size simulated problems, results were indistinguishable from those obtained via an exact, enumerative method. A coarse-scale zoning of Talampaya National Park (Argentina) rendered maps remarkably similar to those produced by subject area experts using a non-quantitative consensus-seeking approach. Results are encouraging and show particular potential for the periodical update of zoning of protected natural areas. Such a capability is crucial for application in developing countries where both human and financial resources are usually scarce but still critical for updating zoning and management plans. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of wavelet transform to hologram analysis: three-dimensional location of particles

The wavelet analysis provides an efficient tool in numerous signal-processing problems and has been implemented in optical processing techniques, such as in-line holography. When the diffraction pattern recorded on a hologram is analyzed by means of a wavelet transform, the 3-D location of small particles can be determined very accurately. The diffraction process can, in fact, be interpreted as a convolution with a family of wavelet functions, or, merely, as a wavelet transform. The scale parameter of the wavelet family is related to the axial distance z that the wave propagates. The original field is then reconstructed by searching for the optimum value of the scale parameter which produces a maximum of the wavelet transform modulus. The technique proposed is implemented and experimental results are presented.     

Comparison between Total Determination and Extractable Heavy Metals from River Sediments using Conventional and Microwave Accelerated Leaching Tests

Abstract

In the present study the concentration of heavy metals (Cu, Cr, Ni, Pb and Zn) in sediments collected from different sampling stations of the Leça river (Portugal) was determined, using Flame Atomic Absorption Spectrometry. In order to estimate the potential mobility of metals in these samples, the results of the total digestion were compared with those obtained by single extractions using EDTA and acetic acid as extractant solutions; in all samples studied, Cu and Zn were found to be the most mobile elements; Ni and Pb showed a smaller mobility in presence of either acidic medium and complexing ligands; Cr was found the least mobilizable element, given that the low extractability obtained with the two extractants tested in this work. In the single extraction tests, microwave energy was also employed to replace the conventional treatment and only in the case of the EDTA the results obtained were similar to those of the conventional procedure (recoveries between 90.16 and 98.76%); the precision (RSD, n=3) of the proposed microwave procedure for EDTA extractions was comparable to those of the conventional method with values always lower than 8% for all metals.     

Development and validation of a HPLC–UV based method for the extraction and quantification of methotrexate in the skin

An innovative and sensitive HPLC–UV method for the extraction and quantification of methotrexate (MTX) in skin layers was developed and validated. Owing to the physico-chemical characteristics of the drug and the nature of the tissue, it was necessary to use folic acid (FA) as an internal standard for MTX quantification in the dermis. MTX (and FA) analysis was performed on a Phenomenex Jupiter C₁₈ column, using a 50 mm sodium acetate buffer (pH 3.6) and methanol mixture (87:13, v/v) as mobile phase, pumped at 1 ml/min. The absorbance was monitored at 290 nm. The method was selective, linear in the range 0.11–8.49 μg/ml for extraction solvent and 0.05–8.94 μg/ml for pH 7.4 phosphate-buffered saline, precise and accurate, with lower limits of quantitation of 0.11 μg/ml (extraction solvent) and 0.05 μg/ml (pH 7.4 phosphate-buffered saline). The method developed is suitable for the quantification of MTX in skin layers at the end of in vitro permeation experiments; the overall mass balance was 96.5 ± 1.4%, in line with the requirements of the Organisation for Economic Co-operation and Development guideline for the testing of the chemicals (Skin absorption: in vitro method).