A Note on a Paper by P. Amodio and F. Mazzia

In 1999 Amodio and Mazzia presented a new backward error analysis for LU factorization and introduced a new growth factor _n . Their very interesting approach allowed them to obtain sharp error bounds. In particular, they derive nice results assuming that partial pivoting is used. However, the forward error bound for the solution of a linear system whose coefficient matrix A is an M-atrix given in Theorem 4.1 of that paper is not correct. They first obtain a bound for the condition number (U) assuming that one has the LU factorization of an M-matrix and then they apply the bounds obtained when partial pivoting is used. But if P is the permutation associated with partial pivoting then PA = LU can fail to be an M-atrix and the bound for (U) can be false, as shown in our Example 1.1. We also prove that, for a pivoting strategy presented in the paper, the growth factor of an M-matrix A is _n(A) = 1 and (U) (A), where U is the upper triangular matrix obtained after applying such a pivoting strategy.This revised version was published online in October 2005 with corrections to the Cover Date.     

A note on the optimal stability of bases of univariate functions

This note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes.     

Identification of acrylate copolymers using pyrolysis and gas chromatography

A combination of pyrolysis and gas chromatography were used to investigate thermal degradation products formed from acrylic copolymers containing alkyl acrylate and methacrylate. The method provided an analytical tool for characterizing the chemical composition and structure of the degradation products. Thermal degradation of the synthesized copolymers was analyzed using isothermal (250 °C) pyrolysis–gas chromatography. The degradation process, and the nature and amount of pyrolysis products, provides relevant information about the thermal degradation of acrylic copolymers and the mechanism of pyrolysis. During pyrolysis, the formation of corresponding olefins, alcohols, acrylates and methacrylate was observed.     

Phosphine and Thiophene Cyclopalladated Complexes: Hydrolysis Reactions in Strong Acidic Media

The mechanisms for the hydrolysis of organopalladium complexes [Pd(CNN)R]BF₄ (R=P(OPh)₃, PPh₃, and SC₄H₈) were investigated at 25 °C by using UV/Vis absorbance measurements in 10 % v/v ethanol/water mixtures containing different sulphuric acid concentrations in the 1.3–11.7 M range. In all cases, a biphasic behavior was observed with rate constants k_1obs, which corresponds to the initial step of the hydrolysis reaction, and k_2obs, where k_1obs>k_2obs. The plots of k_1obs and k_2obs versus sulfuric acid concentration suggest a change in the reaction mechanism. The change with respect to the k_1obs value corresponds to 35 %, 2 %, and 99 % of the protonated complexes for R=PPh₃, P(OPh)₃, and SC₄H₈, respectively. Regarding k_2obs, the change occurred in all cases at about 6.5 M H₂SO₄ and matched up with the results reported for the hydrolysis of the 2‐acetylpyridinephenylhydrazone (CNN) ligand. By using the excess acidity method, the mechanisms were elucidated by carefully looking at the variation of k_i,_obs (i=1,2) versus ${c_{{\rm{H}}^ + } }$ . The rate‐determining constants, k_0,A‐1, k_0,A‐2, and k_0,A‐SE2 were evaluated in all cases. The R=P(OPh)₃ complex was most reactive due to its π‐acid character, which favors the rupture of the trans nitrogen–palladium bond in the A‐2 mechanism and also that of the pyridine nitrogen–palladium bond in the A‐1 mechanism. The organometallic bond exerts no effect on the relative basicity of the complexes, which are strongly reliant on the substituent.     

Evaluation of dispersive liquid-liquid microextraction for the simultaneous determination of chlorophenols and haloanisoles in wines and cork stoppers using gas chromatography-mass spectrometry

Dispersive liquid-liquid microextraction (DLLME) coupled with gas chromatography-mass spectrometry (GC-MS) was evaluated for the simultaneous determination of five chlorophenols and seven haloanisoles in wines and cork stoppers. Parameters, such as the nature and volume of the extracting and disperser solvents, extraction time, salt addition, centrifugation time and sample volume or mass, affecting the DLLME were carefully optimized to extract and preconcentrate chlorophenols, in the form of their acetylated derivatives, and haloanisoles. In this extraction method, 1mL of acetone (disperser solvent) containing 30μL of carbon tetrachloride (extraction solvent) was rapidly injected by a syringe into 5mL of sample solution containing 200μL of acetic anhydride (derivatizing reagent) and 0.5mL of phosphate buffer solution, thereby forming a cloudy solution. After extraction, phase separation was performed by centrifugation, and a volume of 4μL of the sedimented phase was analyzed by GC-MS. The wine samples were directly used for the DLLME extraction (red wines required a 1:1 dilution with water). For cork samples, the target analytes were first extracted with pentane, the solvent was evaporated and the residue reconstituted with acetone before DLLME. The use of an internal standard (2,4-dibromoanisole) notably improved the repeatability of the procedure. Under the optimized conditions, detection limits ranged from 0.004 to 0.108ngmL(-1) in wine samples (24-220pgg(-1) in corks), depending on the compound and the sample analyzed. The enrichment factors for haloanisoles were in the 380-700-fold range.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.