A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type

A time discrete scheme is used to approximate the solution toa phase field system of Penrose–Fife type with a non-conservedorder parameter. An a posteriori error estimate is presentedthat allows the estimation of the difference between continuousand semidiscrete solutions by quantities that can be calculatedfrom the approximation and given data.     

Synthesis and refractive‐index control of fluoropolymers by the polyaddition of bis(epoxide)s and triazine di(aryl ether)s

The polyaddition of fluorine‐containing bis(epoxide)s and fluorine‐containing triazine di(aryl ether)s were examined to give the corresponding fluorine‐containing poly(cyanurate)s. It was observed that the synthesized fluoropolymers had good thermal stabilities and good film‐forming properties. The glass transition temperatures (T_g's) and refractive‐indices (n_D's) of synthesized polymers were determined by differential scanning calorimetry and ellipsometry, respectively, and it was found that the values of T_g's and n_D's were supported by their fluorine containing ratios and skeletons. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4421–4429, 2007     

Performance of various activators in ethylene polymerization based on an iron(II) catalyst system

Ethylisobutylaluminoxane (EBAO) and its analogues were synthesized by a reaction between an triethylaluminum (Et₃Al)/triisobutylaluminum (i‐Bu₃Al) mixture and 4‐fluorobenzeneboronic acid, phenylboronic acid, or n‐butaneboronic acid and subsequent hydrolysis with water. They were used as cocatalysts in ethylene polymerization catalyzed by an iron complex {[(ArN?C(Me))₂C₅H₃N]FeCl₂, where Ar is 2,6‐diisopropylphenyl}. Polyethylene with a high molecular weight and a narrow molecular weight distribution was prepared with modified EBAOs, and the performance of the iron complex at high polymerization temperatures was greatly improved. The activators for the iron complex also affected the polymerization activity and the molecular weight of the resultant polyethylene. It was suggested that the stereo and electronic effects of the substitute groups of aluminoxane contributed to the improved performance of the new activators. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1093–1099, 2004     

Thermoplastic poly(ester‐imide)s derived from anhydride‐terminated polyester prepolymer and diisocyanate

The phase‐separation behavior of thermoplastic poly(ester‐imide) [P(E‐I)] multiblock copolymers, (A‐B)_n, was investigated by a stepwise variation of the imide content. All the multiblock copolymers were synthesized by solution polycondensation with dimethylformamide as a solvent. P(E‐I)s were prepared with anhydride‐terminated polyester prepolymer and diisocyanates. Polyester prepolymers were prepared by the reaction of pyromellitic dianhydride and two different polyols [poly(tetramethylene oxide glycol) (PTMG) and polycaprolactone diol (PCL)]. Structural determination was done with Fourier transform infrared spectroscopy and Fourier transform NMR, and the molecular weight was determined by gel permeation chromatography. The effect of the imide content on the thermal properties of the synthesized P(E‐I)s was investigated by thermogravimetric analysis and differential scanning calorimetry. The polymers were also characterized for static and dynamic mechanical properties. Thermal analysis data indicated that the polymers based on PTMG were stable up to 330 °C in nitrogen atmosphere and exhibited phase‐separated morphology. Polymers based on PCL showed multistage decomposition, and the films derived from them were too fragile to be characterized for static and dynamic mechanical properties. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 341–350, 2004     

Kinetic study of the crosslinking reaction of 1,2‐polybutadiene with dicumyl peroxide in the absence and presence of vinyl acetate

The crosslinking reaction of 1,2-polybutadiene (1,2-PB) with dicumyl peroxide (DCPO) in dioxane was kinetically studied by means of Fourier transform near-infrared spectroscopy (FTNIR). The crosslinking reaction was followed in situ by the monitoring of the disappearance of the pendant vinyl group of 1,2-PB with FTNIR. The initial disappearance rate (R₀) of the vinyl group was expressed by R₀ = k[DCPO]^0.8[vinyl group]^−0.2 (120 °C). The overall activation energy of the reaction was estimated to be 38.3 kcal/mol. The unusual rate equation was explained in terms of the polymerization of the pendant vinyl group as an allyl monomer involving degradative chain transfer to the monomer. The reaction mixture involved electron spin resonance (ESR)-observable polymer radicals, of which the concentration rapidly increased with time owing to a progress of crosslinking after an induction period of 200 min. The crosslinking reaction of 1,2-PB with DCPO was also examined in the presence of vinyl acetate (VAc), which was regarded as a copolymerization of the vinyl group with VAc. The vinyl group of 1,2-PB was found to show a reactivity much higher than 1-octene and 3-methyl-1-hexene as model compounds in the copolymerization with VAc. This unexpectedly high reactivity of the vinyl group suggested that an intramolecular polymerization process proceeds between the pendant vinyl groups located on the same polymer chain, possibly leading to the formation of block-like polymer. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 4437–4447, 2004     

Extension of the chain‐end,free‐volume theory for predicting glass temperature as a function of conversion in hyperbranched polymers obtained through one‐pot approaches

Compared with linear polymers, more factors may affect the glass‐transition temperature (T_g) of a hyperbranched structure, for instance, the contents of end groups, the chemical properties of end groups, branching junctions, and the compactness of a hyperbranched structure. T_g's decrease with increasing content of end‐group free volumes, whereas they increase with increasing polarity of end groups, junction density, or compactness of a hyperbranched structure. However, end‐group free volumes are often a prevailing factor according to the literature. In this work, chain‐end, free‐volume theory was extended for predicting the relations of T_g to conversion (X) and molecular weight (M) in hyperbranched polymers obtained through one‐pot approaches of either polycondensation or self‐condensing vinyl polymerization. The theoretical relations of polymerization degrees to monomer conversions in developing processes of hyperbranched structures reported in the literature were applied in the extended model, and some interesting results were obtained. T_g's of hyperbranched polymers showed a nonlinear relation to reciprocal molecular weight, which differed from the linear relation observed in linear polymers. T_g values decreased with increasing molecular weight in the low‐molecular‐weight range; however, they increased with increasing molecular weight in the high‐molecular‐weight range. T_g values decreased with increasing log M and then turned to a constant value in the high‐molecular‐weight range. The plot of T_g versus 1/M or log M for hyperbranched polymers may exhibit intersecting straight‐line behaviors. The intersection or transition does not result from entanglements that account for such intersections in linear polymers but from a nonlinear feature in hyperbranched polymers according to chain‐end, free‐volume theory. However, the conclusions obtained in this work cannot be extended to dendrimers because after the third generation, the end‐group extents of a dendrimer decrease with molecular weight. Thus, it is very possible for a dendrimer that T_g increases with 1/M before the third generation; however, it decreases with 1/M after the third generation. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1235–1242, 2004     

Thermal behavior of metallodendrimers possessing bis(terpyridinyl)Ru(II) connectivity and different surface functionality

A series of metallodendrimers, assembled by means of bis(terpyridinyl)Ru^(II) connectivity on poly(propylene imine) dendrimer scaffolds, with homogeneous or heterogeneous surfaces, were prepared. Differential scanning calorimetry and thermogravimetric analysis were used to determine their thermal behavior, glass‐transition temperatures, and the decomposition kinetics and temperatures; no synergy effects for these properties were observed for the heterogeneously surfaced constructs in contrast to the corresponding homogeneously coated materials, which exhibited different values depending on their surface functionalities. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1487–1495, 2004     

Segmental and secondary dynamics in hydrogen‐bonded poly(4‐vinylphenol)/poly(methyl methacrylate) blends

Broadband dielectric spectroscopy was used to study the segmental (α) and secondary (β) relaxations in hydrogen‐bonded poly(4‐vinylphenol)/poly(methyl methacrylate) (PVPh/PMMA) blends with PVPh concentrations of 20–80% and at temperatures from ?30 to approximately glass‐transition temperature (T_g) + 80 °C. Miscible blends were obtained by solution casting from methyl ethyl ketone solution, as confirmed by single differential scanning calorimetry T_g and single segmental relaxation process for each blend. The β relaxation of PMMA maintains similar characteristics in blends with PVPh, compared with neat PMMA. Its relaxation time and activation energy are nearly the same in all blends. Furthermore, the dielectric relaxation strength of PMMA β process in the blends is proportional to the concentration of PMMA, suggesting that blending and intermolecular hydrogen bonding do not modify the local intramolecular motion. The α process, however, represents the segmental motions of both components and becomes slower with increasing PVPh concentration because of the higher T_g. This leads to well‐defined α and β relaxations in the blends above the corresponding T_g, which cannot be reliably resolved in neat PMMA without ambiguous curve deconvolution. The PMMA β process still follows an Arrhenius temperature dependence above T_g, but with an activation energy larger than that observed below T_g because of increased relaxation amplitude. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3405–3415, 2004