Effects of uncertainties in the domain on the solution of Dirichlet boundary value problems

Summary. A domain with possibly non-Lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with Lipschitz boundary. A uniquely solvable Dirichlet boundary value problem (DBVP) is defined on each of the Lipschitz domains and the limit of these solutions is investigated. The limit function also solves a DBVP on the limit domain but the problem can depend on the sequences of domains if the limit domain is unstable with respect to the DBVP. The core of the paper consists in estimates of the difference between the respective solutions of the DBVP on two close domains, one of which is Lipschitz and the other can be unstable. Estimates for starshaped as well as rather general domains are derived. Their numerical evaluation is possible and can be done in different ways. Received October 16, 2001 / Revised version received January 16, 2002 / Published online: April 17, 2002 RID="*" ID="*" The research was funded partially by the National Science Foundation under the grants NSF–Czech Rep. INT-9724783 and NSF DMS-9802367 RID="**" ID="**" Support for Jan Chleboun coming from the Grant Agency of the Czech Republic through grant 201/98/0528 is appreciated     

η%-Superconvergence of Finite Element Solutions and Error Estimators

This paper is a summary of our study on the superconvergence of the finite element solutions and error estimators. We will persent the analysis of %-superconvergence for finite element solutions of the Poisson equation in the interior of meshes of triangles with straight edges, as well as the analysis at the boundary. The %-superconvergence via local averaging will also be presented, and the error estimators are compared in the sense of %-superconvergence.     

Optical absorption and photoluminescence studies of Eu-doped phosphate and fluorophosphate glasses

Phosphate (P₂O₅+K₂O+BaO+Al₂O₃+Eu₂O₃) and fluorophosphate (P₂O₅+K₂O+BaO+BaF₂+Al₂O₃+Eu₂O₃) glasses with different Eu³⁺ ion concentrations have been prepared and characterized through optical absorption, photoluminescence and decay times. An intense red luminescence is observed from the ⁵D₀ emitting level of Eu³⁺ ions in these glasses. The relative luminescence intensity ratio of ⁵D₀→⁷F₂→⁵D₀→⁷F₁ transitions has been evaluated to estimate the local site symmetry around the Eu³⁺ ions. The emission spectra of these glasses show a complete removal of degeneracy for the ⁵D₀→⁷F₁ and ⁵D₀→⁷F₂ transitions. Second and fourth rank crystal-field (CF) parameters have been calculated together with the CF strength parameter by assuming the C_2v symmetry for the Eu³⁺ ions in both the phosphate and fluorophosphate glasses. Judd-Ofelt parameters have been evaluated from the luminescence intensity ratios of ⁵D₀→⁷F_J (J=2, 4 and 6) to ⁵D₀→⁷F₁ transitions. These parameters have been used to derive radiative properties such as transition probabilities, branching ratios, radiative lifetimes and peak stimulated emission cross-sections for the ⁵D₀→⁷F_J transitions. Decay curves of the ⁵D₀ level of Eu³⁺ ions in these two Eu³⁺:glass systems have been measured by monitoring the ⁵D₀→⁷F₂ transition (611 nm) at room temperature. The experimental lifetime of the ⁵D₀ level in the title glasses is found to be higher than Eu³⁺-doped niobium phosphate glasses. The analysis indicates that the lifetime of the ⁵D₀ level is found to be less sensitive to the Eu³⁺ ion concentration and addition of BaF₂ has no significant effect on the optical properties of Eu³⁺-doped phosphate glasses.     

Simultaneous estimation of four proton pump inhibitors—lansoprazole,omeprazole, pantoprazole and rabeprazole: development of a novel generic HPLC‐UV method and its application to clinical pharmacokinetic study

A highly selective, sensitive and accurate HPLC method has been developed and validated for the estimation of four proton‐pump inhibitors (PPI), lansoprazole (LPZ), omeprazole (OPZ), pantoprazole (PPZ) and rabeprazole (RPZ), with 500 µL human plasma using zonisamide as an internal standard (IS). The sample preparation involved simple liquid–liquid extraction of LPZ, OPZ, PPZ and RPZ and IS from human plasma with ethyl acetate. The baseline separation of all the peaks was achieved with 0.1% triethylamine (pH 6.0):acetonitrile (72:28, v/v) at a flow rate of 1 mL/min on a Zorbax C₈ column. The total chromatographic run time was 11.0 min and the simultaneous elution of IS, OPZ, RPZ, PPZ and LPZ occurred at approximately 2.42, 4.45, 5.02 and 9.37 min, respectively. The method was proved to be accurate and precise at linearity range of 20.61–1999.79 ng/mL with a correlation coefficient (r) of ≥0.999. The limit of quantitation for each of the PPI studied was 20.61 ng/mL. The intra‐ and inter‐day precision and accuracy values were found to be within the assay variability limits as per the FDA guidelines. The developed assay method was applied to a pharmacokinetic study in human volunteers. Copyright © 2009 John Wiley & Sons, Ltd.     

Fabrication of high performance Gd–Ba–Cu–O single grains in air using a practical melt processing technique

A practical processing route for the fabrication of LRE-Ba–Cu–O single grain superconductors has been developed at the University of Cambridge based on a generic, Mg-doped Nd-123 melt textured seed and suppression of the formation of the solid solution phase in air by enriching the precursors with higher Ba concentration. The processing of high performance Gd–Ba–Cu–O single grains using this processing route is described. The Mg-doped generic seed crystal has been used effectively to promote heterogeneous nucleation via a cold-seeding process. The Gd/Ba solid solution has been suppressed by enriching Gd–Ba–Cu–O precursor powders with two different Ba-rich compositions. This involved adding BaO₂ and GdBa₆Cu₃O_y (Gd-163) (a novel Ba-rich second phase) to the precursor powders, respectively. The Gd-163 phase has been observed not only to suppress formation of the solid solution phase, but also to promote increased heterogeneous grain size. A detailed further study has been carried out with an initial aim of optimizing the BaO₂ and Gd-163 phase content of the precursor composition to produce a single grain almost free of solid solution. Based on the optimized parameters, large single grain Gd–Ba–Cu–O superconductors have been fabricated in an air atmosphere and demonstrated to exhibit record trapped magnetic fields for this material melt processed in air in relatively small single grain samples. The trapped fields of samples produced in air atmosphere are at least comparable to those processed under reduced oxygen partial pressure.     

An Exact Solution for the Optical Absorbance of Thin Films

The electric and magnetic fields within optical thin films are obtained exactly in closed form as solutions of a system of difference equations presented by Berning and utilized by Dill, Neureuther, Tuttle, and Walker in their sample program for modeling optical lithography. The solution allows an explicit closed form expression for the absorbance.     

Aligned magnetic effects through varying permeable bed

The aligned magnetic effects on a steady laminar, viscous, incompressible, conducting fluid down an open inclined channel bounded below by a bed of varying permeability has been studied when the free surface is exposed to atmospheric pressure. Beavers and Joseph slip condition at the interface of the free flow region and the fluid flow in the porous bed and the Darcy’s law in the porous medium have been used. The expressions for velocity, magnetic strength and the mass flow across the cross-section of the channel are obtained.     

Unusual contributions of molecular architecture to rheology and flow birefringence in hyperbranched polystyrene melts

With the increase in sophisticated synthesis methods, it appears that polymer architecture may be a tunable property. Therefore, the role of architecture in rheological and processing properties has received renewed attention, mainly because of dendrimer synthesis and metallocene‐catalyst technology. Linear polymers and hyperbranched polymers represent two ends of branching complexity. Some previous studies have suggested that hyperbranched polymers may behave like unentangled polymers, whereas others have proposed that they exhibit the properties of soft colloids. In an effort to compare the responses of linear and hyperbranched polymers, we synthesized starlike hyperbranched polystyrenes (HBPSs) of various branch lengths and numbers of branches. The HBPSs used in this study were unentangled or weakly entangled, allowing us to study the effect of branch density more readily. Two linear polystyrene (L‐PS) melts and two HBPSs were studied. Using a custom‐built rheooptical apparatus, we characterized the rheology and flow birefringence of these materials. To our knowledge, these are the first flow birefringence measurements on highly branched polymer melts. Our results suggest that the flow behavior of HBPS is significantly different from that of L‐PS: (1) HBPS shows nonterminal behavior in the low‐frequency rheological response; (2) when the stress‐optical rule (SOR) holds, the stress‐optical coefficient of HBPS is much lower than those of analogous linear polymers; and (3) when the branch density is high and the branch length is sufficiently low, the SOR fails for these homopolymer melts. A significant increase in the birefringence for a given amount of stress in the low‐frequency region suggests that there may be a soft core in these materials due to the strong preferential radial orientation of chain segments near the center of a molecule versus those near the periphery. The predominantly elastic response of the soft structures may be responsible for the enhanced form birefringence. Our preliminary results indicate that these materials may exhibit both polymeric and soft‐colloid natures. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2562–2571, 2001     

Effect of substrate bias voltage on the structure, electric and dielectric properties of TiO2 thin films by DC magnetron sputtering

Titanium dioxide (TiO₂) films have been deposited on glass and p-silicon (1 0 0) substrates by DC magnetron sputtering technique to investigate their structural, electrical and optical properties. The surface composition of the TiO₂ films has been analyzed by X-ray photoelectron spectroscopy. The TiO₂ films formed on unbiased substrates were amorphous. Application of negative bias voltage to the substrate transformed the amorphous TiO₂ into polycrystalline as confirmed by Raman spectroscopic studies. Thin film capacitors with configuration of Al/TiO₂/p-Si have been fabricated. The leakage current density of unbiased films was 1 × 10⁻⁶ A/cm² at a gate bias voltage of 1.5 V and it was decreased to 1.41 × 10⁻⁷ A/cm² with the increase of substrate bias voltage to −150 V owing to the increase in thickness of interfacial layer of SiO₂. Dielectric properties and AC electrical conductivity of the films were studied at various frequencies for unbiased and biased at −150 V. The capacitance at 1 MHz for unbiased films was 2.42 × 10⁻¹⁰ F and it increased to 5.8 × 10⁻¹⁰ F in the films formed at substrate bias voltage of −150 V. Dielectric constant of TiO₂ films were calculated from capacitance–voltage measurements at 1 MHz frequency. The dielectric constant of unbiased films was 6.2 while those formed at −150 V it increased to 19. The optical band gap of the films decreased from 3.50 to 3.42 eV with the increase of substrate bias voltage from 0 to −150 V.