On the homogenization of some linear problems in domains weakly connected by a system of traps

The aim of the paper is to study the asymptotic behaviour of solutions of second‐order elliptic and parabolic equations, arising in modelling of flow in cavernous porous media, in a domain Ω^ε weakly connected by a system of traps ??^ε, where ε is the parameter that characterizes the scale of the microstructure. Namely, we consider a strongly perforated domain Ω^ε ?Ω a bounded open set of ?³ such that Ω^ε =Ω₁^ε ∪Ω₂^ε ∪??^ε ∪ W^ε, where Ω₁^ε, Ω₂^ε are non‐intersecting subdomains strongly connected with respect to Ω, ??^ε is a system of traps and meas W^ε → 0 as ε → 0. Without any periodicity assumption, for a large range of perforated media and by means of variational homogenization, we find the homogenized models. The effective coefficients are described in terms of local energy characteristics of the domain Ω^ε associated with the problem under consideration. The resulting homogenized problem in the parabolic case is a vector model with memory terms. An example is presented to illustrate the methodology. Copyright © 2007 John Wiley & Sons, Ltd.     

Novel Biodegradable Poly(pentadecalactone-co-oxo-crown ether) Studied with Solid-State 1H and 13C NMR

Two grades of a novel biodegradable copolymer of ω-pentadecalactone (PDL) and 2-oxo-12-crown-4 (OC), with respective molar compositions of 77/23 and 52/48, were characterized with 1D and 2D MAS ¹H and ¹³C NMR spectroscopy. The results indicate that both copolymers are semi-crystalline with PDL divided over the crystalline and amorphous phase, and OC exclusively located in the amorphous phase. Proton T₂ and T_1ρ relaxation confirm the existence of small crystalline domains.     

Identifying the Role of Brønsted and Lewis Acid Sites in the Diels-Alder Cycloaddition of 2,5-DMF and Ethylene

The role of Lewis and Brønsted acid sites in the Diels-Alder cycloaddition (DAC) of ethylene to 2,5-dimethylfuran (2,5-DMF) to p-xylene was investigated. Amorphous silica catalysts containing Al³⁺ (ASA), Ga³⁺ (ASG), and In³⁺ (ASI) were prepared via homogeneous deposition-precipitation. Silica modified with Zr⁴⁺ (ASZ) was prepared by impregnation. Their acidic properties were characterized by various IR and NMR spectroscopic techniques. Measurements using pyridine as a probe molecule highlighted the presence of mostly Lewis acid sites (LAS) in all materials. Using CO as a probe, in contrast, demonstrated the existence of Brønsted acid sites (BAS) in ASA and ASG, which were nearly absent in ASI and ASZ. Differences in basic strength can explain the contrast in results observed between the two probe molecules. The highest p-xylene yield (~20 %) in the DAC reaction, could be achieved with ASA and ASG. The lack of BAS in ASI and ASZ resulted in inferior performance in the DAC, with p-xylene yields below 5 %. These results indicate the importance of BAS for the DAC reaction. Several other heterogeneous and homogeneous catalysts were explored for the DAC reaction to show the generality of our conclusion that BAS play a critical role in obtaining p-xylene from 2,5-DMF and ethylene.     

Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature

The connection between orthogonal polynomials, Padé approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Padé and Padé-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5. This revised version was published online in June 2006 with corrections to the Cover Date.     

Direct Measurement of Electron Beam Scattering in the Low Vacuum SEM

The beam-gas interactions in the low-vacuum SEM is investigated through experimental and simulation approaches. An original experimental way to characterise the skirting is proposed. We obtained directly image distribution of the primary electron beam and the scattered electrons in the low-vacuum SEM. The experimental results are compared to the electron beam profile obtained with Monte-Carlo simulation.     

KZn(HP2O7)·2H2O and KMn(HP2O7)·2H2O: acid pyrophosphate metallates(II) with layer structures

The crystal structures of the isomorphous title compounds, namely potassium zinc hydrogen pyrophosphate dihydrate and potassium manganese hydrogen pyrophosphate dihydrate, consist of acidic pyrophosphate–metallate(II) layers joined by K⁺ ions and hydrogen‐bridging bonds. The Zn²⁺/Mn²⁺ ions are octahedrally surrounded by four pyrophosphate O atoms and by two water mol­ecules. The (HP₂O₇)^3? anions exhibit eclipsed conformations. The metal ions and water O atoms lie on mirror planes, as does the central O atom of the (HP₂O₇)^3? anion.     

Fast atom bombardment mass spectrometric study of alkali metal cationized unsaturated C-glycosides

Fast atom bombardment mass spectrometry of alkali metal cationized unsaturated C-glycosides is a useful method for the determination of the molecular masses of these compounds. The mass-analyzed ion kinetic energy and collision-activated dissociation mass-analyzed ion kinetic energy spectra of the pairs of anomers are distinguishable and are characteristic of each anomer.     

Structural evolutions of the n-heneicosane and n-tricosane molecular alloys at 293 K

A structural study of odd-numbered n-alkane (C_n) binary mixtures (C₂₁ : C₂₃) was carried out on powder samples using a Guinier-de Wolff camera with increasing concentration of n-C₂₃ at 293 K.

Despite the reports in the literature, these molecular alloys do not form an orthorhombic continuous homogeneous solid solution to C₂₁ from C₂₃ at “low temperature”. Instead, as already observed in two even-numbered C_n systems, X-ray diffraction results show the existence of seven solid solutions as the molar concentration of C₂₃ increases: four terminal solid solutions, denoted β₀(C₂₁)β₀(C₂₃), isostructural with the “low temperature” phase of pure C₂₁ and C₂₃ (Pbcm), β′₀(C₂₁) and β′₀(C₂₃), identical to the phase β′₀ which appears in pure C₂₃ above the δ transition, and three orthorhombic intermediate solid solutions, designated β″₁, β′₁ and β″₂.

On the basis of powder X-ray photographs, the phases β″₁ and β″₂ (C₂₁ : C₂₃) are indistinguishable, and they are isostructural with the intermediate solid solution β″ of the even-numbered C_n binary systems (C₂₂ : C₂₄) and (C₂₄ : C₂₆). The phase β′₁(C₂₁ : C₂₃) is also isostructural with the two indistinguishable intermediate solid solutions β′₁ and β′₂ of the molecular alloys (C₂₂ : C₂₄) and (₂₄ : C₂₆).

From this study and our other laboratory results, the sequences of appearance of the solid solutions and the structural identities between these phases are established at “low temperature” for all the binary molecular alloys of consecutive C_n (odd-odd, even-even or odd-even: 19 < n < 27) when increasing the solute concentration.