Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type. Communicated by Vincent Rivasseau Submitted: October 28, 2005; Accepted: February 17, 2006     

Des équations de Dirac et de Schrödinger pour la transformation de Fourier

Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrödinger equation on the half-axis and has used, in tandem, the Gel'fand–Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms. To cite this article: J.-F. Burnol, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

New example of a non-heme mononuclear iron(IV) oxo complex. Spectroscopic data and oxidation activity

The green complex S=1 [(TPEN)FeO]2+ [TPEN=N,N,N',N'-tetrakis(2-pyridylmethyl)ethane-1,2-diamine] has been obtained by treating the [(TPEN)Fe]2+ precursor with meta-chloroperoxybenzoic acid (m-CPBA). This high-valent complex belongs to the emerging family of synthetic models of Fe(IV)=O intermediates invoked during the catalytic cycle of biological systems. This complex exhibits spectroscopic characteristics that are similar to those of other models reported recently with a similar amine/pyridine environment. Thanks to its relative stability, vibrational data in solution have been obtained by Fourier transform infrared. A comparison of the Fe=O and Fe=(18)O wavenumbers reveals that the Fe-oxo vibration is not a pure one. The ability of the green complex to oxidize small organic molecules has been studied. Mixtures of oxygenated products derived from two- or four-electron oxidations are obtained. The reactivity of this [FeO]2+ complex is then not straightforward, and different mechanisms may be involved.     

Polyelectrolyte multilayers and degradable polymer layers as multicompartment films

Polyelectrolyte multilayers are now a well established concept with numerous potential applications in particular as biomaterial coatings. To timely control the biological activity of cells in contact with a substrate, multicompartment films made of different polyelectrolyte multilayers deposited sequentially on the solid substrate constitute a promising new approach. In a first paper (Langmuir 2004, 20, 7298) we showed that such multicompartment films can be designed by alternating exponentially growing polyelectrolyte multilayers acting as reservoirs and linearly growing ones acting as barriers. In the present study, we first demonstrate however that these barriers composed of synthetic polyelectrolytes are not degraded despite the presence of phagocytic cells. We propose an alternative approach where exponentially growing poly(L-lysine)/hyaluronic acid (PLL/HA) multilayers, used as reservoirs, are alternated with biodegradable polymer layers consisting in poly(lactic-co-glycolic acid) (PLGA) and acting as barriers for PLL chains that diffuse within the PLL/HA reservoirs. We first show that these PLGA layers can be deposited alternatively with PLL/HA multilayers leading to polyelectrolyte multilayer/hydrolyzable polymeric layer films and acting as a reservoirs/barriers system. Bone marrow cells seeded on these films ending by a PLL/HA reservoir rapidly degrade it and internalize the PLL chains confined in this reservoir. Then the cells degraded locally the PLGA barrier and internalize the PLL localized in a lower (PLL/HA) compartment after 5 days of seeding. By changing the thickness of the PLGA layer, we hope to be able to tune the time delay of degradation. Such mixed architectures made of polyelectrolyte multilayers and hydrolyzable polymeric layers could act as coatings allowing us to induce a time scheduled cascade of biological activities. We are currently working on the use of comparable films with compartments filled by proteins or peptides and in which the degradation of the barriers results from a hydrolysis over tunable time scales.     

Layer by layer buildup of polysaccharide films: physical chemistry and cellular adhesion aspects

The formation ofpolysaccharide films based on the alternate deposition of chitosan (CHI) and hyaluronan (HA) was investigated by several techniques. The multilayer buildup takes place in two stages: during the first stage, the surface is covered by isolated islets that grow and coalesce as the construction goes on. After several deposition steps, a continuous film is formed and the second stage of the buildup process takes place. The whole process is characterized by an exponential increase of the mass and thickness of the film with the number of deposition steps. This exponential growth mechanism is related to the ability of the polycation to diffuse "in" and "out" of the whole film at each deposition step. Using confocal laser microscopy and fluorescently labeled CHI, we show that such a diffusion behavior, already observed with poly(L-lysine) as a polycation, is also found with CHI, a polycation presenting a large persistence length. We also analyze the effect of the molecular weight (MW) of the diffusing polyelectrolyte (CHI) on the buildup process and observe a faster growth for low MW chitosan. The influence of the salt concentration during buildup is also investigated. Whereas the CHI/HA films grow rapidly at high salt concentration (0.15 M NaCl) with the formation of a uniform film after only a few deposition steps, it is very difficult to build the film at 10(-4) M NaCl. In this latter case, the deposited mass increases linearly with the number of deposition steps and the first deposition stage, where the surface is covered by islets, lasts at least up to 50 bilayer deposition steps. However, even at these low salt concentrations and in the islet configuration, CHI chains seem to diffuse in and out of the CHI/HA complexes. The linear mass increase of the film with the number of deposition steps despite the CHI diffusion is explained by a partial redissolution of the CHI/HA complexes forming the film during different steps of the buildup process. Finally, the uniform films built at high salt concentrations were also found to be chondrocyte resistant and, more interestingly, bacterial resistant. Therefore, the (CHI/HA) films may be used as an antimicrobial coating.     

Volume and heat capacity of model non-aqueous self-assembly systems

Blends of immiscible polymers are often stabilized by block copolymers which can form non-aqueous micelles and microemulsions in the liquid polymers. The phase diagrans, apparent volumes and apparent heat capacities of model non-aqueous binary and ternary systems were studied in order to investigate the conditions under which such self-assembly systems could form. 1,2-Hexanediol, which can cosolubilize hexane and ethyleneglycol, forms inverse micelles in hexane and weak microaggregates in ethyleneglycol. Genapol X-060, a commercial alcoholic surfactant containing on the average an aliphatic chain of 13 carbons and 6 oxyethylenes (C₁₃E₆), forms microaggregates in poly(ethyleneglycol) 400. These self-assembly systems are strengthen in the presence of a third component which has an affinity for the inner phase.Presented at the Symposium, 76^th CSC Congress, Sherbrooke, Quebec, May 30–June 3, 1993, honoring Professor Donald Patterson on the occasion of his 65^th birthday.     

A New Method to Prepare Transition Metal Salts of Bulk and Supported Heteropolyacids. Application to the Catalysis of the Oxidative Dehydrogenation of Isobutyric Acid

Transition metal salts of heteropolyacids have been prepared taking into account the strong acidic and cation exchanging properties of the solid heteropolyacids. The exchange between protons and the transition metal cation is carried out by stirring a suspension of the hydrated heteropolyacid in a solution of the metal acetylacetonate complex in toluene. The exchange occurs on the surface of the solid particles and diffusion of protons and metal cations into the hydrated lattice leads to the substitution of all the protons. The method can be utilized in order to prepare supported vanadyl and copper molybdophosphates from supported heteropolyacids and they have been studied in the catalysis of the oxidative dehydrogenation of isobutyric acid. The effect of vanadyl counter-ions on the catalytic behavior is the same as observed with bulk catalysts but, on the contrary, copper supported molybdophosphate shows an acid catalytic activity not observed with bulk catalysts.     

Hydrogen activation on Mo-based sulfide catalysts,a periodic DFT study

Hydrogen adsorption on Mo[bond]S, Co[bond]Mo[bond]S, and Ni[bond]Mo[bond]S (10 1 macro 0) surfaces has been modeled by means of periodic DFT calculations taking into account the gaseous surrounding of these catalysts in working conditions. On the stable Mo[bond]S surface, only six-fold coordinated Mo cations are present, whereas substitution by Co or Ni leads to the creation of stable coordinatively unsaturated sites. On the stable MoS(2) surface, hydrogen dissociation is always endothermic and presents a high activation barrier. On Co[bond]Mo[bond]S surfaces, the ability to dissociate H(2) depends on the nature of the metal atom and the sulfur coordination environment. As an adsorption center, Co strongly favors molecular hydrogen activation as compared to the Mo atoms. Co also increases the ability of its sulfur atom ligands to bind hydrogen. Investigation of surface acidity using ammonia as a probe molecule confirms the crucial role of sulfur basicity on hydrogen activation on these surfaces. As a result, Co[bond]Mo[bond]S surfaces present Co[bond]S sites for which the dissociation of hydrogen is exothermic and weakly activated. On Ni[bond]Mo[bond]S surfaces, Ni[bond]S pairs are not stable and do not provide for an efficient way for hydrogen activation. These theoretical results are in good agreement with recent experimental studies of H(2)[bond]D(2) exchange reactions.