Reiter's condition (P2) and hypergroup representations

Let K be a hypergroup with Haar measure. We relate Reiter's condition $(P_2)$ to positive definite measures and associated representations, induced representations from a subgroup of K and convolution operators on $L_2(K)$. To cite this article: L. Pavel, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

We consider finite difference approximations of solutions of inverse Sturm‐Liouville problems in bounded intervals. Using three‐point finite difference schemes, we discretize the equations on so‐called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm‐Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand‐Levitan formulation. © 2005 Wiley Periodicals, Inc.     

Activity coefficients of water and sulfolane from the solid + liquid phase diagram

The solid + liquid phase diagram drawn from thermal analysis is presented. It exhibits a eutectic at x₂ = 0.303 and T = 262.2 K and precludes the existence of definite compounds between the two components. Mixed crystals are observed in the sulfolane-rich region. The activity coefficients of water, f_{x, 1}, and of sulfolane, f_{x, 2}, are calculated from the initial freezing points of water. Log f_{x, 1} and log f_{x, 2} show a steady dependence on volume composition, providing evidence for the supposition that strongly self-associated water and monomer sulfolane scarcely interact with one another, and that the thermodynamic properties of the mixtures should rather be related to steric factors.     

Reversible phase transitions in emulsified nanostructured lipid systems

Aqueous submicron-sized dispersions of the binary monolinolein/water system, which are stabilized by means of a polymer, internally possess a distinct nanostructure. Taking this as our starting point, we were able to demonstrate for the first time that the internal structure of the dispersed particles can be tuned by temperature in a reversible way. Upon increasing the temperature, the internal structure undergoes a transition from cubic via hexagonal to fluid isotropic, the so-called L2 phase, and vice versa. Intriguingly, in addition to the structural changes in topology, the particles expel (take up) water to (from) the aqueous continuous phase when increasing (decreasing) the temperature in a reversible way. At each temperature, the internal structure of the dispersed particles corresponds very well to the structure observed in nondispersed binary monolinolein with excess water. This agreement is independent of any thermal history (including phase transitions), which proves that the structures in the dispersed particles actually are in thermodynamic equilibrium with the surrounding water phase.     

Structural diversity and similar bioactivity in synthetic bicyclononanes

Simple syntheses of diverse bicyclo[3.3.1]nonanes and related compounds as the minimal substructure of bioactive natural products via Michael, aldol, and alkylation reactions from diketones are described herein. The structures of the synthesized compounds were determined by infrared spectroscopy, NMR (¹H and ¹³C), and electrospray ionization–high-resolution mass spectrometry. We also show the in vitro antimicrobial activity against Gram-positive and Gram-negative bacteria. The qualitative analysis has revealed that the new synthesized compounds 5, 6, 9, and 11 present antibacterial properties.     

Striking Features of DPH Electronic Spectra as a Function of Multicomponent Solvent Nature

1,6-Diphenyl-1,3,5-hexatriene is one of the most used fluorescent labels in cell membrane fluidity studies. However, its solubility in aqueous biological media is low and an initial solubilizing step is necessary. In this study, the authors present spectroscopic and computational results in order to explain the striking displacement observed in electronic spectra (absorption and fluorescence) in different solvents. The experimental results were correlated with those obtained by molecular dynamics simulations of multicomponent solutions, which revealed non-homogenous microscopic regions formation in the ternary aqueous DPH solutions.