Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2

On arbitrary polygonal domains $\Omega \subset \RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(\Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s \in (2,\frac{5}{2})$ to $s \in (1,\frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned.     

Molecular dynamics simulations of the penetration lengths: application within the fluctuation theory for diffusion coefficients

Mutual diffusion in condensed phases is a theoretically and practically important subject of active research. One of the most rigorous and theoretically advanced approaches to the problem is a recently developed approach based on the concept of penetration lengths (Physica A 320 (2003) 211; Physica A 322 (2004) 151). In the current study, a fast molecular dynamics scheme has been developed to determine the values of the penetration lengths in Lennard–Jones binary systems. Results deduced from computations provide a new insight into the concept of penetration lengths. It is shown for four different binary liquid mixtures of non-polar components that computed penetration lengths, for various temperatures and compositions, are consistent with those deduced from experiments in the framework of the formalism of the fluctuation theory. Moreover, the mutual diffusion coefficients obtained from a coupled fluctuation theory and molecular dynamics scheme exhibit consistent trends and average deviations from experimental data around 10–20%.     

Rossby Waves on a Shear Flow with Recirculation Cores

Large-amplitude Rossby waves riding on a background flow with a weak shear can be calculated up to a critical amplitude for which the meridional velocity, in a frame traveling with the wave, approaches zero at some point. Here we consider waves with an amplitude slightly greater than the critical amplitude by incorporating a region of recirculating fluid (vortex core) near this critical point. The effect of the vortex core is to introduce an extra nonlinear term into the equation for the wave amplitude proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. The main effect due to the vortex core is a broadening of the wave profile. Furthermore, we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.     

Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problemsEstimations de Carleman globales pour des solutions faibles de problèmes elliptiques avec condition de Dirichlet non homogène

We consider a general second order elliptic equation with right-hand side $\text{f+∑}{}_{\mathrm{j=0}}{}^{\mathrm{N}}\text{?f}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{j}}\text{∈}\text{H}{}^{\mathrm{?1}}\text{(Ω)}$ where $\text{f,f}{}_{\mathrm{j}}\text{∈}\text{L}{}^2\text{(Ω)}$ and Dirichlet boundary condition g∈H^1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L² norms of f and f_j and the H^1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.     

Conformational flexibility of the 1,4-dihydropyridine ring in calcium channel agonists and antagonists molecules

Conformational flexibility of the 1,4-dihydropyridine ring in 4-aryl derivatives of the 2,6-dimethyl-3,5-dicarbmethoxy-1,4-dihydropyridine has been studied using the MM3 molecular mechanics method. Change of the C=C---C---C= endocyclic torsion angle of dihydrocycle in the range of 35° entails an increase of energy less than 1 kcal mol⁻¹. The energy levels below 99% of the molecular population for each molecule were estimated. The interval of the torsion angle change for these regions is about 60°.