Reaction of epoxy resin and hyperbranched polyacids

The condensation reaction between two different epoxy resins and a hyperbranched polyester (MAHP) [poly(allyloxy maleic acid‐co‐maleic anhydride)] was studied. We compared two kinds of diglycidyl ether bisphenol A type of epoxy resins with different molecular weights, that is, epoxy resin GY240 (M = 365 g/mol) and GT6064 (M = 1540 g/mol) in this reaction. The results showed a marked difference in their reaction pattern in terms of ability to form crosslinked polymer networks with MAHP. For the former low‐molecular‐weight epoxy resin, no crosslinking could be observed in good solvents such as THF or dioxane within the set of reaction conditions used in this study. Instead, polymers with epoxide functional degrees between 0.34 and 0.5 were formed. By contrast, the latter high‐molecular‐weight epoxy resin, GT6064, rapidly produced highly crosslinked materials with MAHP under the same reaction conditions. The spherical‐shape model of hyperbranched polymer was applied to explain this difference in reaction behavior. Hence, we have postulated that low‐molecular‐weight epoxy resins such as GY240 are unable to crosslink the comparatively much bigger spherically shaped MAHP molecules. However, using high‐molecular‐weight epoxy resins greatly enhances the probability of crosslinking in this system. Computer simulations verified the spherical shape and condensed bond density of MAHP in good solvents, and submicron particle analysis showed that the average MAHP particle size was 9 nm in THF. Furthermore, the epoxy‐functionalized polyesters were characterized by ¹H NMR and FTIR, and the molecular weights and molecular‐weight distributions were determined by size‐exclusion chromatography. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 4457–4465, 2000     

Matrix stretching for sparse least squares problems

For linear least squares problems min _x ‖Ax − b‖₂, where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to catastrophic fill in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and a more general splitting method. We show that for both schemes the stretched problem has the same set of solutions as the original least squares problem. Further, the condition number of the stretched problem differs from that of the original by only a modest factor, and hence the approach is numerically stable. Experimental results from applying the recursive binary scheme to a set of modified matrices from the Harwell‐Boeing collection are given. We conclude that when A has a small number of dense rows relative to its dimension, there is a significant gain in sparsity of the factor R. A crude estimate of the optimal number of splits is obtained by analysing a simple model problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behavior of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with a recursive algorithm for computing the coefficients in the expansions. An application to perturbed epidemic SIS models is discussed.