Densité des vecteurs propres généralisés d'une classe d'opérateurs compacts non auto-adjoints et applications

We consider a closed densely defined linear operatorT in a Hilbert spaceE, and assume the existence of ₀ (T) such thatK = (T - ₀ I)^-1 is compact and the existence ofp>0 such thats _n (K)=o((n ^–1/p)), whereS _n (K) denotes the sequence of non-zero eigenvalues of the compact hermitian operator . In this work, sufficient conditions (announced in [1]) are introduced to assure that the closed subspace ofE spanned by the generalized eigenvectors ofT coincides withE. These conditions are in particular verified by a family of non-self-adjoint operators arising in reggeon's field theory.     

Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans la théorie des champs de reggeons

In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory. The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator: $$\begin{gathered} H_{\lambda ',\mu ,\lambda ,\alpha } = \lambda '\sum\limits_{j = 1}^N {A_j^{ * 2} A_j^2 + \mu \sum\limits_{j = 1}^N {A_j^ * A_j + i\lambda \sum\limits_{j = 1}^N {A_j^ * (A_j + A_j^ * )A_j } } } \hfill \\ + \alpha \sum\limits_{j = 1}^{N - 1} {(A_{j + 1}^ * A_j + A_j^ * A_{j + 1} ),} \hfill \\ \end{gathered}$$ whereA* _j andA _j ,j∈[1,N] are the creation and annihilation operators. In the third section, we restrict our study to the case of nul transverse dimension (N=1). Following the study done in [1], we consider the operator: $$H_{\lambda ',\mu ,\lambda } = \lambda 'A^{ * 2} A^2 + \mu A^ * A + i\lambda A^ * (A + A^ * )A,$$ whereA* andA are the creation and annihilation operators. For λ′>0 and λ′²≦μλ′+λ². We prove that the solutions of the equationu′(t)+H _{λ′, μ,λ} u(t)=0 are expandable in series of the eigenvectors ofH _λ′,μ,λ fort>0. In the last section, we show that the smallest eigenvalue σ(α) of the operatorH _{λ′,μ,λ,α} is analytic in α, and thus admits an expansion: σ(α)=σ₀+ασ₁+α²σ₂+..., where σ₀ is the smallest eigenvalue of the operatorH _{λ′,μ,λ,0}.     

Cycloaddition reactions in aqueous systems: A two-decade trend endeavor

Cycloaddition reactions are an integral and weighty part of organic chemistry in pedagogy and research as well. The wealthy literature on cycloaddition reactions from their birth up to now, unequivocally witnesses to their leading chemistry. The so-called “conventional solvents” are organic solvents that have indubitably promoted their success. Yet, the toxicity facet of these solvents impedes their use freely and with no fear. Not only is the operating chemist uncomfortable while experimenting, but also the environment is equally threatened. Working out the cycloaddition reactions and other organic ones in aqueous systems would certainly bring some relief to the chemist and to the environment as well. Unusual outcomes in terms of yield, reactivity and selectivity compared to those performed in organic solvents were commonly observed, and have overwhelmed the chemists with surprise indeed. In this review, homo Diels–Alder reactions in aqueous media include those involving the following dienophiles: maleimides, α,β-unsaturated esters, p-benzoquinones, vinyl ketones, phenyl-1-(2-pyridyl)-2-propen-1-one, α,β-unsaturated esters. A special case is the organocatalysis of Diels–Alder cycloaddition of α,β-unsaturated ketones (aldehydes). Of no less importance, some hetero Diels–Alder cycloaddition reactions in water systems are delineated. The impact of additives (salts, organic and inorganic chemicals), micellar catalysis and Lewis/Brønstëd acid catalysis on outcomes of such cycloaddition reactions is discussed. The 1,3-dipolar cycloaddition methodology applied to aqueous media has brought forth a number of heterocyclic compounds, usually with a regio- and stereoselectivity pecularity. These heterocycles include triazoles, tetrazoles, pyrazoles, isoxazoles, isoxazolidines, pyrroles and pyrrolidines. The superiority of copper(I) catalysis in the azide-alkyne cycloaddition (Huisgen cycloaddition) in water is endorsed by a number of examples.     

Facile synthesis of Cu-LDH with different Cu/Al molar ratios: application as antibacterial inhibitors

Research on Chemical Intermediates - A range of Cu-LDHs has been synthesized by co-precipitation using metal nitrate precursors and sodium carbonate under varying molar ratios Cu/Al...     

Assessment of the intrinsic interactions of mesoporous silica with carbon dioxide

Three mesoporous silica, SBA-16, SBA-15 and MCM-41, with different structures and porosities were synthesized via a hydrothermal method and their interactions with carbon dioxide (CO₂) were investigated through thermal programmed desorption (TPD) and differential scanning calorimetry. TPD measurements provided precise assessments of the intrinsic affinity towards CO₂, without the influence of moisture. All silica materials were found to exhibit intrinsic affinity towards carbon dioxide, but the surface basicity, expressed in terms of retained CO₂ amount, is markedly influenced by increases in pore size and framework structures. SBA-15 displayed the highest CRC values, explained in terms of larger pore size, lower numbers of acidic out-of plane Si–OH and higher numbers of much less acidic in-plane silanols. These findings provide valuable information for a better understanding of the role of the silica structure in the intrinsic basicity, prior to further modifications for improving the affinity towards CO₂ or merely for catalysis purposes involving CO₂ as reagents, intermediates or products.     

Surface vibrational thermodynamics from ab initio calculations for fcc(1 0 0)

We present vibrational dynamics and thermodynamics for the (1 0 0) surfaces of Cu, Ag, Pd, Pt and Au using a real space approach. The force field for these systems is described by density functional theory. The changes in the vibrational dynamics and thermodynamics from those in bulk are confined mostly to the first-layer. A substantial enhancement of the low-frequency end of the acoustic branch was found and is related to a loosening of the bond at the surface. The thermodynamics of the first-layer also show significant differences (higher heat capacity, lower free energy and higher mean vibrational square amplitudes) from what obtains in bulk. Comparing these results with those calculated using embedded-atom method potentials, we discovered that for Ag(1 0 0) and Cu(1 0 0), the two methods yield very similar results while for Pd(1 0 0), Pt(1 0 0) and Au(1 0 0) there are substantial differences.