Bounds for the torsional rigidity of orthotropic cylinders-additional results

Résumé Une inegalité est derivée et utilisée pour améliorer des limites pour la rigidité torsionelle d'une cylindre orthotropique obtenu dans une étude antérieure.     

Bounds for the torsional rigidity of an isotropic cylinder with no end warping

We obtain upper bounds for the torsional rigidity of an isotropic right cylinder whose ends are restrained against warping in terms of the Saint-Venant torsional rigidity, the polar moment of inertia and the lowest free membrane eigenvalue of the cross-section, the length of the cylinder and the elastic constants. They may be used to show that the two torsional rigidities tend to coincidence as the cylinder becomes infinitely long. Various other implications of the bounds are also discussed.     

Nonlinear stability for a thermofluid in a vertical porous slab

A novel Liapunov functional was used in previous work to establish nonlinear stability of certain nontrivial equilibrium states; essentially the context was that of pure nonlinear diffusion. This paper uses the same Liapunov functional to derive a nonlinear stability criterion in the context of a highly nonlinear system of p.d.e.'s involving nonlinear diffusion as an element. The context is that of convection of a thermofluid (i) conforming to Darcy's law and the Boussinesq approximation, (ii) with temperature dependent thermal diffusivity and viscosity, in an infinite vertical slab of porous material. The vertical faces are held at different constant temperatures, a steady state is identified, and is shown to be nonlinearly stable provided that the Rayleigh number does not exceed a quantity which reflects the temperature dependence of the pertinent physical properties. It may be that the versatility of the Liapunov functional thus exhibited may extend to other nonlinear systems involving nonlinear diffusion. Received September 4, 1998     

Moving boundary i.b.v.p.s for Schrödinger equations: Solution bounds and related results

The paper is concerned with i.b.v.p.s for Schrödinger equations, linear and nonlinear, in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the L₂ norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region. The application of the methodology to i.b.v.p.s for other p.d.e.s is discussed briefly.     

Imprinted nanomaterials: a new class of synthetic receptors

The molecular imprinting approach provides a unique opportunity for the creation of three-dimensional cavities with tailored recognition properties. Over the last decade this field has expanded considerably, across a variety of disciplines, leading to novel approaches and many potential applications. Progress in the field of materials science has led to significant breakthroughs and the application of the imprinting approach to novel polymeric formats offers new insights and attractive methods for the preparation of synthetic receptors. In particular, nanomaterials have received considerable attention in the developing field of nanotechnology. With a large number of recent developments in the field of molecular imprinting available, this article is focused on a selection of new systems, in particular the different formats of nanomaterials, such as nanogels, nanofibres, nanowires and nanotubes.     

Bounds for torsional rigidity with no end warping

Summary We obtain an upper bound for the torsional rigidity of an isotropic right cylinder when there is no end warping in terms of the Saint Venant torsional rigidity, the polar moment of inertia and the lowest free membrane eigenvalue of the cross-section, the length, of the cylinder and the elastic constants. It may be used to show that the two torsional rigidities tend to coincidence as the cylinder becomes infinitely long.

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Résumé Nous obtenons une borne supérieure pour la rigidité à la torsion d'un cylindre droit isotrope dont chaque extrémité est maintenue dans un plan rigide; cette borne s'exprime à l'aide de la rigidité à la torsion de Saint-Venant, du moment polaire d'inertie de la section et de la valeur propre fondamentale de la membrane libre, ainsi que de la longueur du cylindre et des constantes d'élasticité. Elle peut servir à démontrer que ces deux rigidités coïncident quand la longueur du cylindre tend vers l'infini.

     

Upper Estimates for a Moving Boundary Problem for Resonant Nonlinear Schrödinger Equations

Pointwise bounds are obtained for the solution of an initial boundary value problem for the resonant nonlinear Schrödinger equations. The context is that of a straight-line region with prescribed moving boundaries, expanding or noncontracting, upon which zero (Dirichlet) conditions are imposed.     

Preparation of New Wittig Reagents and Their Application to the Synthesis of alpha,beta-Unsaturated Phosphonates

The Wittig reagent [(diethoxyphosphinyl)methylidene]triphenylphosphorane (1b) has been successfully synthesized for the first time via its phosphonium triflate salt (4a), by treating (diethoxyphosphinyl)methyl triflate with triphenylphosphine. The procedure has been applied to the synthesis of other phosphoranes and phosphonium salts. The new Wittig reagents thus synthesized were treated with various aldehydes and an activated ketone, affording the corresponding alpha,beta-unsaturated phosphonates. Triphenylphosphorane 1b and triphenylphosphonium 4a led to both cis and trans isomers with the latter being predominant, while trans isomers were almost exclusively formed when tributyl reagents (1c and 4d) were used.     

The use of FTIR and NMR spectroscopies to study prepolymerisation interactions in nitrogen heterocycles

A detailed investigation into the functional groups responsible for the formation of a noncovalent complex between 2-aminopyridine (template) and methacrylic acid (functional monomer) has been carried out using FTIR spectroscopy and confirmed by ¹H NMR spectroscopic data. The approach adopted to confirm the mechanism of interaction was the analysis of the template plus the structurally similar 2-methylaminopyridine and 2-dimethylaminopyridine. A 1:1 stoichiometry of complexation was determined by Job plot analysis following titration, with FTIR results complementing those of the ¹H NMR study. The strength of interaction between 2-aminopyridine and the functional monomer measured through band shifts by FTIR spectroscopy was compared with such interactions for the isomers 3- and 4-aminopyridine. This comparison identified a clear correlation between template pK _a, degree of interaction and subsequent nonspecific binding in the nonimprinted polymer. Using FTIR spectroscopy it was also possible to observe the effect of temperature on the prepolymerisation solution. IR spectra showed that lower temperatures led to more stabilized interactions of the hydrogen-bonded complex. The potential advantages of FTIR spectroscopy compared with ¹H NMR spectroscopy in studying prepolymerisation solutions have been identified.