Global existence and stability for an inverse coefficient problem for a semilinear parabolic equation

We establish the global existence, the uniqueness and the stability for an inverse coefficient problem for a semilinear parabolic equation. This problem is motivated by an application related to laser material treatments, and our result is a continuation of the previous work by Hömberg and Yamamoto (Inverse Problems 22:1855–1867, 2006). We transform our inverse problem into a nonlinear integral equation of second kind, which is solved locally by the contraction mapping principle. Next we prove that the maximal solution of the integral equation is a global solution.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

Structural study by X-ray diffraction, magnetic susceptibility and Mössbauer spectroscopy of the Na2Mn2(1 − x)Cd2xFe(PO4)3 (0 ≤ x ≤ 1) solid solution

Na₂Mn_{2(1 − x)}Cd_2xFe(PO₄)₃ (0 ≤ x ≤ 1) phosphates were prepared by solid state reaction and characterized by powder X-ray diffraction, magnetic susceptibility and Mössbauer spectroscopy. The X-ray diffraction patterns indicated the formation of a continuous solid solution which crystallizes in the alluaudite structural type characterized by the general formula X(2)X(1)M(1)M(2)₂(PO₄)₃. The cation distribution, deduced from a structure refinement of the x = 0, 0.5 and 1 compositions, is ordered in the X(2) sites and disordered in the remaining X(1), M(1) and M(2) sites. The magnetic susceptibility study revealed an antiferromagnetic behaviour of the studied compounds. The ⁵⁷Fe Mössbauer spectroscopy confirmed the structural results and proved the exclusive presence of Fe³⁺ ions.     

Synthesis of bridgehead-substituted azabicyclo[2.2.1]heptane and -[3.3.1]nonane derivatives for the elaboration of α7 nicotinic ligands

Azabicyclo[2.2.1]heptane and -[3.3.1]nonane scaffolds (X = Cl, Br) containing a pyridinyl substituent at the bridgehead position were prepared via two complementary chemical pathways, either by the transformation of a methoxy group into a synthetically valuable chlorine atom at the C-6 position of the pyridine moiety or by means of a regioselective C-6 deprotonation/halogenation process of the pyridine moiety exemplified by chlorination or bromination. These newly generated scaffolds were then engaged in Suzuki-Miyaura coupling reactions to provide α7 nicotinic ligands. Both chemical series were evaluated in vitro for their affinity at α7 nicotinic receptors, revealing nanomolar potency with significant selectivity over the α4β2 nicotinic subtype. These approaches offer a general access to these α7 nicotinic scaffolds and ligands.     

Screening method for the evaluation of asymmetric catalysts for the reduction of aliphatic ketones

ATH reductions of aliphatic ketones in water catalyzed by ruthenium coordinated by prolinamide ligands produce alcohols with moderate enantiomeric excesses in most cases. A set of seven aliphatic ketones is proposed for a rapid evaluation of the enantioselectivity of catalysts by one-pot multi-substrates reduction. The screening of a library of prolinamides shows that according to the structure of the ketones different ligands give the best asymmetric inductions.     

Intervalence charge transfer in mixed valence compound modified by the formation of a supramolecular complex

The electrochemical and spectroelectrochemical properties of N,N-diphenyl-1,4-phenylenediamine (PDA) were investigated in the absence and in the presence of 18-crown-6-ether (18C6) or dibenzo 24-crown-8-ether (DB24C8), in a solution of tetrabutylammonium hexafluorophosphate (TBAPF6) in acetonitrile and in the presence of trifluoroacetic acid (TFA) only for 18C6. In neutral acetonitrile, PDA undergoes two reversible oxidation processes, which lead first to the formation of the cation-radical considered as mixed valence (MV) compound, and then to the dicationic species. When 18C6 is added in the medium and depending on 18C6 concentration, cyclic voltammetry shows a marked shift to more cathodic potentials of the current waves of the second redox process only. This is attributed to a strong interaction between the PDA(+2) dication and two 18C6 molecules, leading to the formation of a supramolecular complex with an association constant value K(a) = 7.0 × 10(7) M(-2). The interaction of 18C6 with PDA(+2) dication has a direct effect on the PDA(+.) cation-radical corresponding to a decrease in the lifetime of the MV compound and of the intramolecular electron transfer rate when 18C6 is present. Indeed, it results in a large decrease in the intervalence charge transfer (IV-CT) between the two amine centers in the MV compound (k(th) = 1.35 × 10(10) s(-1) in 18C6-free neutral solution containing 5.0 × 10(-4) M PDA, and k(th) = 3.6 × 10(9) s(-1) in the same medium at [18C6]/[PDA] = 20/1). And the comproportionation constant K(co) falls from 6.0 × 10(6) in 18C6-free solution to 1.6 × 10(3) at [18C6]/[PDA] = 20/1. In acidified acetonitrile and when TFA concentration is increased, PDA still shows the two successive and reversible oxidation processes, but both are shifted to more anodic potentials. However, when 18C6 is added, the two oxidation waves shift to more cathodic potentials, indicating an interaction of all protonated PDA redox states with 18C6, resulting in the formation of supramolecular complexes. In the presence of TFA, the value of K(co) is decreased to 4.3 × 10(4), but it remains unchanged when 18C6 is added, indicating no change in the lifetime of the MV compound. In this medium, IV-CT in the MV compound is greater with 18C6 (k(th) = 2.3 × 10(10) s(-1) for [18C6]/[PDA] = 20/1) than without (k(th) = 1.4 × 10(9) s(-1)), which indicates a more important IV-CT rate when 18C6 is present. The results show for the first time that is it possible to control the IV-CT rate, through the lifetime and the potential range where the MV compound is the most important. This control is not obtained as usual by chemical modification of the structure of the starting molecule, but by varying either the acidity or the 18C6 concentration as external stimuli, which lead to reversible formation/dissociation of a supramolecular complex species. Moreover, we also studied the electrochemical properties of PDA in the presence of wider crown ether such as DB24C8. We showed that PDA undergoes the same electrochemical behavior with DB24C8 than with 18C6 in neutral organic medium (K(a) = 2.9 × 10(3) M(-1)). This result suggests that the complexation between the electrogenerated PDA(+2) dication and the crown ethers may occur through face-to-face mode rather than rotaxane mode even with DB24C8 which is supposed to form inclusion complexes.     

Pyochelin, a siderophore of Pseudomonas aeruginosa: physicochemical characterization of the iron(III), copper(II) and zinc(II) complexes

Pseudomonas aeruginosa is an opportunistic pathogen, synthesizing two major siderophores, pyoverdine (Pvd) and pyochelin (Pch), to cover its needs in iron(III). If the high affinity and specificity of Pvd toward iron(III) (pFe = 27.0) was well described in the literature, the physicochemical and coordination properties of Pch toward biologically relevant metals (Fe(III), Cu(II) or Zn(II)) have been only scarcely investigated. We report a thorough physico-chemical investigation of Pch (potentiometry, spectrophotometries, ESI/MS) that highlighted its moderate but significantly higher affinity for Fe(3+) (pFe = 16.0 at p[H] 7.4) than reported previously. We also demonstrated that Pch strongly chelates divalent metals such as Zn(II) (pZn = 11.8 at p[H] 7.4) and Cu(II) (pCu = 14.9 at p[H] 7.4) and forms predominantly 1 : 2 (M(2+)/Pch) complexes. Kinetic studies revealed that the formation of the ferric Pch complexes proceeds through a Eigen-Wilkins dissociative ligand interchange mechanism involving two protonated species of Pch and the Fe(OH)(2+) species of Fe(III). Our physico-chemical parameters supports the previous biochemical studies which proposed that siderophores are not only devoted to iron(III) shuttling but most likely display other specific biological role in the subtle metals homeostasis in microorganisms. This work also represents a step toward deciphering the role of siderophores throughout evolution.     

Stability for the Multi-Dimensional Borg-Levinson Theorem with Partial Spectral Data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues λ _k and the normal derivatives ?φ _k /?ν of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to O(k ^?α) with α ? 1, then we still have Hölder stability.     

On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.