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.     

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.     

Lacitemzine, a novel sesquiterpene acid from the Tunisian plant Pulicaria laciniata (Coss. et Kral.) Thell.

The flowers of Pulicaria laciniata (Coss. et Kral.) Thell. (Asteraceae) afforded a new sesquiterpene acid 1, named lacitemzine together with the three known compounds, 4-hydroxy-3-methoxypyridine 2, β-sitosterol-3-O-β-d-glucoside 3 and 1,3,5-trimethoxybenzene 4. The structure of compound 1, 2-(2,6-dimethyl-3,4,5,6,7,8,9,10-octahydro-5,8-oxaazulen-9-yl)acrylic acid, contains a guaiane skeleton and was elucidated by spectroscopic procedures including 2D-NMR and X-ray diffraction.     

New antioxidant bibenzyl derivative and isoflavonoid from the Tunisian Salsola tetrandra Folsk

Two new phytochemical compounds, Tetranins A and B, 1-(3,5'-dihydroxy-4'-méthoxyphenyl)-2-phenylethanol 1 and 5,2'-dihydroxy-5'-methoxy-6,7-methylenedioxy-isoflavone 2, were isolated from the EtOAc extract of Salsola tetrandra roots. They exhibited a significant antioxidant effect in 1,1-diphenyl-2-picrylhydrazyl (DPPH) free radical and 2,2'-azinobis (3-ethylbenzothiazoline-6-sulphonic acid (ABTS) assays. Their structures were elucidated by extensive spectroscopic methods including 1-D-((1)H and (13)C) and 2-D-NMR experiments (CHcorr, HMBC, (1)H-(1)H COSY and NOESY) as well as high-resolution ES-MS and they were found to be quite active as antioxidants in the DPPH and ABTS assays.     

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.     

Inhibition of Fungi and Gram-Negative Bacteria by Bacteriocin BacTN635 Produced by Lactobacillus plantarum sp. TN635

The aim of this study was to evaluate 54 lactic acid bacteria (LAB) strains isolated from meat, fermented vegetables and dairy products for their capacity to produce antimicrobial activities against several bacteria and fungi. The strain designed TN635 has been selected for advanced studies. The supernatant culture of this strain inhibits the growth of all tested pathogenic including the four Gram-negative bacteria (Salmonella enterica ATCC43972, Pseudomonas aeruginosa ATCC 49189, Hafnia sp. and Serratia sp.) and the pathogenic fungus Candida tropicalis R2 CIP203. Based on the nucleotide sequence of the 16S rRNA gene of the strain TN635 (1,540 pb accession no FN252881) and the phylogenetic analysis, we propose the assignment of our new isolate bacterium as Lactobacillus plantarum sp. TN635 strain. Its antimicrobial compound was determined as a proteinaceous substance, stable to heat and to treatment with surfactants and organic solvents. Highest antimicrobial activity was found between pH 3 and 11 with an optimum at pH?=?7. The BacTN635 was purified to homogeneity by a four-step protocol involving ammonium sulfate precipitation, centrifugal microconcentrators with a 10-kDa membrane cutoff, gel filtration Sephadex G-25, and C18 reverse-phase HPLC. SDS-PAGE analysis of the purified BacTN635, revealed a single band with an estimated molecular mass of approximately 4 kDa. The maximum bacteriocin production (5,000 AU/ml) was recorded after a 16-h incubation in Man, Rogosa, and Sharpe (MRS) medium at 30?°C. The mode of action of the partial purified BacTN635 was identified as bactericidal against Listeria ivanovii BUG 496 and as fungistatic against C. tropicalis R2 CIP203.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

On the prescribed scalar curvature problem on Sn: The degree zero case

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere $S^n$, $n?3$. We give new existence and multiplicity results based on a new Euler–Hopf formula type. Our argument also has the advantage of extending the well known results due to Y. Li (1995) [10].     

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.     

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.