On the Normality of the Sum of Two Normal Operators

The aim of this paper is to give sufficient conditions on two normal operators (bounded or not), defined on a Hilbert space, which make their algebraic sum normal. The results are accompanied by some interesting examples and counter examples.     

Kinetic and equilibrium factors affecting saturation of chromium oxide in soda-silicate melts

Factors controlling the rate at which chromium oxide reaches saturation in Na₂O-xSiO₂ liquids have been studied as a function of melt composition and oxygen fugacity. Under an oxidizing atmosphere, liquid Na₂CrO₄ or Cr₂O₃ crystals can be in equilibrium with soda-silicate melts, depending on the concentration of sodium in the studied system. Under reducing conditions, NaCrSi₂O₆ is stabilized in silica-rich melts when T is lower than 1160 °C, while Cr₂O₃ is in equilibrium with the sodium-rich melts when T is above 1160 °C. The chromium oxide (Cr₂O₃) to pyroxene (NaCrSi₂O₆) transformation is described in terms of the time required to reach chemical and textural equilibrium. Na₂CrO₄, NaCrSi₂O₆, and Cr₂O₃ phase stability domains are reported as well as the Na₂O-SiO₂-Cr₂O₃ phase diagram in the studied temperature and fO₂ range.     

Rearrangement inequalities in the discrete setting and some applications

We prove extended Hardy–Littlewood, Riesz and Pólya–Szeg? inequalities in the discrete case. We also establish cases of equality in these inqualities and give some applications of our result.     

Conformal metrics with prescribed boundary mean curvature on balls

We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler–Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.     

Quantum Noise and Squeezed Light by Dipolaritons in the Nonlinear Regime

The quantum noise and the squeezing effect, in the transmitted light, by a dipolariton system formed by a double quantum wells microcavity in the strong coupling regime are investigated. It is shown that the indirect exciton nonlinearity generates stronger squeezing than the direct exciton nonlinearity, nevertheless produces very high fluctuations for some particular detunings and nonlinearities. The system shows a considerable sensitivity to the thermal excitations where the coupling to the direct and indirect excitonic thermal baths progressively destroys the nonclassical effect. Despite that, indirect exciton squeezing manifests a higher resistance against the thermal bath temperature. As a result, the emission field is strongly governed by the indirect excitons, varying between robust squeezing and excess noise.     

Minimizers of a Class of Constrained Vectorial Variational Problems: Part I.

In this paper, we prove the existence of minimizers of a class of multiconstrained variational problems. We consider systems involving a nonlinearity that does not satisfy compactness, monotonicity, neither symmetry properties. Our approach hinges on the concentration-compactness approach. In the second part, we will treat orthogonal constrained problems for another class of integrands using density matrices method.     

Computation of the spectra of some integral operators and application to the numerical solution of some linear integral equations

In this paper, we present two methods of computing the spectrum of a compact integral operator. The first method is based on an exact matrix representation of the operator. The second method uses a convenient quadrature method to discretisize the integral operator and to provide accurate approximations to the spectrum and the eigenfunctions of this later. Also, we show how our methods can be used in the framework of some stable procedures for the approximation of f^† the normal solution of the minimal L²-norm of the integral equation of the first kind Af = g, which is often an ill-posed equation. These procedures are based on a spectral expansion of the operator A. To finish, we give some numerical examples that illustrate the results of this work.     

Commutativity up to a factor for bounded and unbounded operators

In this paper, we further investigate the problem of commutativity up to a factor (or λ-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to the problem. We finish the paper by a conjecture on the commutativity of self-adjoint operators.     

Unbounded operators having self-adjoint,subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A² is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\ge 2$, then T is normal. Hence a closed subnormal operator T such that $T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that $A^p$ and $A^q$ are self-adjoint for some coprime numbers p and q, then A must be self-adjoint.