The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators

The Lie theoretic nature of the Rankin-Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL(2,R)-equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl(2,C) Verma modules. The Rankin-Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin-Cohen type brackets for Hilbert modular forms. Then, some Rankin-Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL(2,R) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin-Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.     

Large deviations estimates for self-intersection local times for simple random walk in {\mathbb{Z}}^3

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.      

Sphalerons on orbifolds

In this work, we study the electroweak sphalerons in a 5D background, where the fifth dimension lies on an interval. We consider two specific cases: flat space-time and the anti-de Sitter space-time compactified on S ¹/Z ₂. In our work, we take the SU(2) gauge–Higgs model, where the gauge fields reside in the 5D bulk; but the Higgs doublet is confined in one brane. We find that the results in this model are close to those of the 4D Standard Model (SM). The existence of the warp effect, as well as the heaviness of the gauge Kaluza–Klein modes make the results extremely close to the SM ones.     

Results of Vidav-Palmer Type in m-Convex Algebras

Using the notion of a normal cone, we give results of Vidav-Palmer type in terms of hermitian, quasi-hermitian and uniformly quasi-hermitian elements. Consequences are also given.     

Cu(II) adsorption by halloysites intercalated with sodium acetate

Intercalated halloysites with sodium acetate at various contact time were prepared. The resulting materials were characterised by X-ray powder diffraction, Fourier transformed infrared spectroscopy, scanning electronic microscopy, and specific surface area evaluation. The modified halloysites were employed as Cu(II) adsorbents from aqueous solutions. Various parameters were studied through the batch method. Kinetic data, equilibrium isotherms, and thermodynamic parameters were evaluated by considering several models. The fraction of halloysite intercalated with sodium acetate remained low, up to 7 days, then linearly increased with contact time. Thirty days were required to achieve a ratio of 91%. For this sample, a proliferation of small tubes (nanotubes) was evidenced. The best results were achieved with the pseudo-second-order kinetic model associated with intraparticle diffusion and with the Redlich-Peterson isotherm, for the equilibrium data. The thermodynamic data show that adsorption would be spontaneous at low temperatures, of exothermic nature, resulting in an adsorbate-adsorbent system much more ordered. The insertion of CH(3)COONa into halloysite significantly affects the Cu(II) adsorption. The magnitude in enhancement of copper adsorption on solid phase thus depends on the content of the carboxylic functional groups, which increases with the insertion of CH(3)COONa into the halloysitic matrix. The involved mechanism is quite complex. It implies electrostatic considerations and a cationic exchange process. The most intercalated sample was found to be very effective as adsorbent of copper(II) from aqueous solutions.     

Ring opening polymerization of tetrahydrofuran catalysed by maghnite-H+

The cationic ring-opening polymerization of tetrahydrofiiran using maghnite-H~+ is reported.Maghnite-H~+,is a non-toxic solid catalyst issued from proton exchanged montmorillonite clay.Polytetrahydrofuran,also called "poly(butandiol) ether",with acetate and hydroxyl end groups was successfully synthesized.Effects of reaction temperature, weight ratio of initiator/monomer and reaction time on the conversion of monomer and on the molecular weight are investigated.A cationic mechanism of the reaction was proposed.This chemistry can be considered as a suitable route for preparing poly(THF) as a soft segment for thermoplastic elastomers.     

<Superscript>99m</Superscript>Tc-Labeled human and camel lactoferrin for detection of <Emphasis Type="Italic">Staphylococcus aureus</Emphasis> infections

Human and camel lactoferrin, were labeled with technetium-99m using a simple technique that yielded a labelling efficiency of 98% and stable complexes over time. The camel Lactoferrin was purified. The proteins were investigated for targeting infections in experimental animals. The biodistribution of those labeled proteins in mice infected with Staphylococcus aureus at the femoral muscle was investigated and test ratios of 3.43 and 3.71 were obtained after 60 min for the camel and the human Lactoferrin, respectively. These values were compared to each other and to those reported in the literature, leading us to conclude that the radiotracers are efficient.     

Identifying the Structural Evolution of the Sodium Ion Battery Na2FePO4F Cathode

Na₂FePO₄F is a promising cathode material for Na‐ion batteries owing to its relatively high discharge voltage and excellent cycling performance. Now, the long‐ and short‐range structural evolution of Na₂FePO₄F during cycling is studied by in situ high‐energy X‐ray diffraction (XRD), ex situ solid‐state nuclear magnetic resonance (NMR), and first‐principles DFT calculations. DFT calculations suggest that the intermediate phase, Na_1.5FePO₄F, adopts the space group of P2₁/c, which is a subgroup (P2₁/b11, No. 14) of Pbcn (No. 60), the space group of the starting phase, Na₂FePO₄F, and this space group provides a good fit to the experimental XRD and NMR results. The two crystallographically unique Na sites in the structure of Na₂FePO₄F behave differently during cycling, where the Na ions on the Na2 site are electrochemically active while those on the Na1 site are inert. This study determines the structural evolution and the electrochemical reaction mechanisms of Na₂FePO₄F in a Na‐ion battery.     

Characterization of mediated and non-mediated oxidase enzyme based glassy carbon electrodes

The performance and analytical characteristics of a glassy carbon glutaraldehyde immobilized glucose oxidase electrode have been established with regard to the direct detection of hydrogen peroxide produced from the reaction of glucose with oxygen. Measurements were performed at + 1.1 V vs. SCE, and selectivity was obtained by casting the surface with a cellulose acetate membrane. Results compared favorably with the classical platinum-enzyme probe. The mechanism of ascorbic acid interference in hydrogen peroxide detection is reported. Mediated detection was also investigated for oxidase enzymes (glucose oxidase and xanthine oxidase) immobilized on the bare glassy carbon electrode. The probes were characterized using a specific enzyme mediator in solution (phenazine methosulfate or dichlorophenol-indophenol) plus hexacyanoferrate(III) as an electrochemical mediator. The electrode was poised at + 0.36 V vs. SCE for the detection of hexacyanoferrate(II). The advantages of this dual mediator configuration include high stability and sensitivity of the electrochemical signal and the ability to use less positive potentials for increased selectivity. Application to other enzymes, such as hydrogenases, using such a binary redox configuration is suggested.