Global Behavior of Positive Solutions of a Generalized Lane–Emden System of Nonlinear Differential Equations

In this paper we discuss the existence and the global behavior of positive solutions of the following generalized Lane–Emden system of differential equations:

$$\begin{aligned} -u''= & {} a(x)u^{\alpha }\,v^{r}\quad \text{ in } (0,1), \\ -v''= & {} b(x)u^{s}\,v^{\beta }\quad \, \text{ in } (0,1), \\ u'(0)= & {} v'(0)=0; \quad \, u(1)=v(1)=0, \end{aligned}$$

where \(r,\,s\in {\mathbb {R}}\), \(\alpha ,\,\beta <1\) such that \(\gamma :=(1-\alpha )(1-\beta )-rs>0\) and the nonnegative functions \(a,\,b\) satisfy some conditions related to the Karamata regular variation theory.

     

An optimisation method for separating and rebuilding one-dimensional dispersive waves from multi-point measurements. Application to elastic or viscoelastic bars

When using a classical SHPB (split Hopkinson pressure bar) set-up, the useful measuring time is limited by the length of the bars, so that the maximum strain which can be measured in material testing applications is also limited. In this paper, a new method with no time limits is presented for measuring the force and displacement at any station on a bar from strain or velocity measurements performed at various places on the bar. The method takes the wave dispersion into account, as must inevitably be done when making long time measurements. It can be applied to one-dimensional and single-mode waves of all kinds propagating through a medium (flexural waves in beams, acoustic waves in wave guides, etc.). With bars of usual sizes, the measuring time can be up to 50 times longer than the time available with classical methods. An analysis of the sensitivity of the results to the accuracy of the experimental data and to the quality of the wave propagation modelling was also carried out. Experimental results are given which show the efficiency of the method.     

Identification de la relation de dispersion dans les barresIdentification of the dispersion relation in rods

Dispersion and attenuation of longitudinal waves in elastic or weakly viscoelastic rods are measured by analysing the resonant frequencies present in the strain spectrum due to an unknown loading. The method takes the finite measuring time of the test into account. It is applied to an aluminium bar, in which the dispersion relation is identified very accurately at frequencies up to 60 kHz. To cite this article: R. Othman et al., C. R. Mecanique 330 (2002) 849–855.     

The Aldehyde Ketone Rearrangement. Rearrangement of Bis(4-methoxyphenyl)-acetaldehyde into 4,4′-Dimethoxydeoxybenzoin

The formyl carbonyl group in bis(4-methoxyphenyl)-acetaldehyde does not contribute to the formation of the carbonyl group in 4,4′-dimethoxydeoxybenzoin when the former rearranges to the latter by treatment with 50% (w/w) sulfuric acid.     

Combined perturbation effects for a class of nonlinear fourth-order Navier problems

We study a class of nonlinear elliptic problems with Navier boundary condition and involving the Laplace and the biharmonic operators. The main result of this paper establishes a sufficient condition for the existence of nontrivial weak solutions, in relationship with the values of a certain real parameter with respect to the principal eigenvalue of the Laplace operator.     

Thermal stability and UV–Vis-NIR spectroscopy of a new erbium-doped fluorotellurite glass

A new transparent bulk glass from the system 76TeO₂?·?10ZnO?·?9.0PbO?·?1.0PbF₂?·?3.0Na₂O doped with Er³⁺ (TZPPN doped with Er³⁺) has been prepared using the conventional melt-quenching method. Results of differential thermal analysis (DTA) measurements indicate good thermal stability of this glass. The refractive indices at different wavelengths, the optical energy gap, the Sellmeier gap energy and the dispersion energy have been estimated. The Judd–Ofelt parameters, Ω _t (t?=?2,?4,?6) of Er³⁺ were evaluated from optical absorption spectra. Electric dipole, magnetic dipole type transition probabilities, spectroscopic quality factors, branching ratio and radiative lifetimes of several excited states of Er³⁺ have been predicted using intensity Judd–Ofelt parameters. The spectroscopic properties indicate that TZPPN glass doped with Er³⁺ is a promising candidate for laser applications and may be suitable for upconversion fibre optical devices.     

Multigrid solvers in reconfigurable hardware

The problem of finding the solution of partial differential equations (PDEs) plays a central role in modeling real world problems. Over the past years, Multigrid solvers have showed their robustness over other techniques, due to its high convergence rate which is independent of the problem size. For this reason, many attempts for exploiting the inherent parallelism of Multigrid have been made to achieve the desired efficiency and scalability of the method. Yet, most efforts fail in this respect due to many factors (time, resources) governed by software implementations. In this paper, we present a hardware implementation of the V-cycle Multigrid method for finding the solution of a 2D-Poisson equation. We use Handel-C to implement our hardware design, which we map onto available field programmable gate arrays (FPGAs). We analyze the implementation performance using the FPGA vendor's tools. We demonstrate the robustness of Multigrid over other similar iterative solvers, such as Jacobi and successive over relaxation (SOR  ), in both hardware and software. We compare our findings with a C++$\mathrm{C}++$ version of each algorithm. The obtained results show better performance when compared to existing software versions.     

Ternary complexes in gels from agarose and from chemically‐modified agarose

Thermodynamic data and mechanical measurements are shown for gels prepared in aqueous binary solvents (water/DMSO, water/DMF, water/methyl formamide and water/formamide). When electrostatic interactions, as opposed to hydrogen bonding, can be established with the cosolvent (DMSO, DMF, methyl formamide) we come to the conclusion that ternary complexes are formed (agarose/water/cosolvent). In the case of chemically‐modified agarose (OH groups replaced by OCH₃ groups) we suggest that these cosolvents are directly involved in the formation of the gel.     

Transient electric birefringence study of agarose aggregates in water-dimethyl sulphoxide mixtures

The dynamical behavior of agarose aggregates in a mixture of water and dimethyl sulphoxide (DMSO) has been investigated using a Transient Electric Birefringence (TEB) set up. Different agarose concentrations for different water/DMSO proportions were considered. The relaxation of the birefringence curves can be fitted with a stretched exponential of the form Δn(t) = Δn0 exp(-(Dt)^0.5). The exponent 0.5 suggests a high polydispersity of the aggregates. The rotational diffusion coefficient, D, depends strongly on the water/DMSO proportion. For small values of the applied electric field it presents a maximum for a water volume fraction equal to 0.5. A strong dependence of D on the applied electric field is also observed. However, the same kind of behavior as for weak fields is recovered when a subtle representation of the data is used. Differential Scanning Calorimetry (DSC) results also show a maximum in the temperature of fusion of the aggregates for a water volume fraction of 0.5. This set of results suggest strongly that « complexes » of water/DMSO/agarose are formed in the mixture and that the process of aggregate formation is more « efficient » for a water/DMSO mixture of same proportions.