Rigged modules I: Modules over dual operator algebras and the Picard group

In a previous paper we generalized the theory of $W^{?}$-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak^?-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel “Rigged modules II”, giving many new results about weak^?-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty }(\mathbb{D})$, and prove that for a weak* closed function algebra A, the weak Picard group is a semidirect product of the automorphism group of A, and the subgroup consisting of symmetric equivalence bimodules.     

On the distribution of cumulative Parisian ruin

We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyse for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.     

A note on the convexity of ruin probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.     

Supercritical carbon dioxide extraction and gas chromatography–mass spectrometry analysis of Gymnema sylvestre R.Br. roots

Supercritical fluid extraction (SFE) achieves a wide range of applications since the past decade as a sustainable green technology. The present study investigates the process for producing high yield by supercritical carbon dioxide (Sc-CO₂) extraction form Gymnema sylvestre R.Br. roots. The effect of temperature and pressure on the percentage of accumulative yield is demonstrated. It is found that the highest yield is obtained at the temperature of 60°C and the pressure of 10?MPa. A proper review showed that there is a lack in the study of Sc-CO₂ extraction of this plant especially for optimization of SFE process which makes this study useful and valuable. For more benefit, the extract which achieves highest percentage is subjected to gas chromatography–mass spectrometry to study its chemical composition and detect the active principle compounds which present with high concentration and expected to be responsible of the pharmaceutical properties of the extract.     

Isolation of berberine tannate using a chromatography activity-guided fractionation from root bark of Berberis hispanica Boiss. & Reut.

Root bark of Berberis hispanica Boiss. & Reut. constitutes an important source of alkaloids and is traditionally used in Algeria and Morocco in many herbal formulations particularly for the treatment of stomach infections and colon cancer. Activity-guided fractionation of Berberis extract is developed by step-gradient elution on column chromatography followed by preparative HPLC to isolate the most antistaphylococcal compound. Solvent extraction-activity screening indicates that ethanol should be used as the extracting solvent as it has exhibited the highest activity against Staphylococcus aureus(Rosenbach ATCC6538) followed by water extract, whereas no activity is recorded for acetone and hexane extracts. Screening of collected fractions from chromatographic separations indicates that the fraction presenting the highest antistaphylococcal activity is identified as berberine tannate by UV spectrum, melting point measurement and confirmed by liquid chromatography coupled to mass spectrometry (LC/MS). The minimal inhibition concentration value of berberine tannate against S. aureus is evaluated at 5?µg?mL^?1.