Theorems of Stein-Rosenberg type. III. The singular case

In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J_ω and the successive overrelaxation (SOR) matrix

, in the case when the Jacobi matrix J₁ is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J_ω and

have been established which do not depend on the assumption that J₁ is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix A with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that J_ω and

simultaneously semiconverge for all ω in the real interval [0,1).     

Fluorescence study of the interaction between metal ions and methyl methacrylate–methacrylic acid copolymers in aqueous solutions: thallium(I), calcium(II), and terbium(III)

 The binding of the cations thallium(I), calcium(II) and terbium(III) to methyl methacrylate– methacrylic acid copolymers with different fractions of acid groups (x) has been studied in aqueous solution at various pH values using the fluorescence of covalently bonded 9-vinyl anthracene as a probe. In all cases, the extent of binding increases as a function of the charge of the polymer with either increasing fraction of carboxylic acids or of pH. However, differences are observed in the behavior of the three cations. With Tl(I), quenching of the anthracene group fluorescence is observed, indicating that the thallium(I) approaches the probe and suggesting that the alkylanthracene is probably in a relatively polar region. Binding constants have been determined from anthracene quenching data and from studies with the fluo-rescent-probe sodium pyrenetetra-sulfonate. Good agreement is obtained between the two methods, and values for the binding constants increase from 250 to 950 M^-1 as x increases from 0.39 to 1. It is suggested that the cation is held in the polyelectrolyte domain, partly by Debye–Hückel effects and partly by more specific interactions. Stronger binding is found with calcium(II) and terbium(III), and in this case increases in fluorescence intensity are observed on complexation due to the anthracene group being in a more hydrophobic region, probably as a result of conformational changes in the polymer chain. In the former case the stoichiometry of the interaction was determined from the fluorescence data to involve two carboxylate groups bound per calcium. Association constants were found using murexide as an indicator of free calcium to vary from 8400 to 37 000 M^-1 as x increases from 0.39 to 1. It is suggested that in this case specific calcium(II)–carboxylate interactions contribute to the binding. With terbium(III), a greater increase in the probe fluorescence intensity was observed than with calcium, and it is suggested that the interaction with the polymer is even stronger, leading to a more pronounced conformational change in the polymer. It is proposed that the terbium(III) interacts with six carboxylic groups on the polymer chain, with three being coordinated and three attracted by electrostatic interactions. Received: 10 June 1997 Accepted: 24 October 1997     

THE INTERACTION OF THE EXCITED STATES OF SAFRANINE T WITH ALIPHATIC AMINES IN ORGANIC SOLVENTS

Abstract—
The reactions of the excited states of safranine T with aliphatic amines have been studied in methanol and acetonitrile. Quenching of the singlet and triplet states occurs by different mechanisms. Whereas the former excited state is quenched by a charge-transfer mechanism, the triplet state is quenched through proton transfer from the excited dye to the amine. This process leads to the unprotonated form of the dye in the triplet state, which is later quenched by amines to form the corresponding semireduced species. The monoprotonated triplet also undergoes self-quenching in both solvents (k = 1.2 × 10⁸ M ^-1 s^-1).