Hilbert Space of Probability Density Functions Based on Aitchison Geometry

The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.     

Minor components with the γ-cyclogeranil geraniol skeleton from Bellardia trixago (L.) all.

Trixagol 1 and fourteen derivatives with the skeleton of γ-cyclogeranil geraniol were isolated from Bellardia trixago, the structures were determined by spectroscopic methods and confirmed by partial synthesis. Furthermore. 3,4-dihydro-γ-ionone, α-ambrinol, β-sitosterol and three flavonoids 5-O-demethyl tangeretin, 5-hydroxy auranetin and 3′-methoxy calycopterin were also isolated.     

Derivative spectrophotometric resolution of mixtures of the food colourants Tartrazine, Amaranth and Curcumin in a micellar medium

A study of the absorption spectra of the food colourants Tartrazine (E-102), Amaranth (E-123) and Curcumin (E-100) in various organic solvents and pH media, have been carried out. In addition, the spectral behaviour in aqueous micellar systems of anionic, cationic and non-ionic surfactants, as well as beta-cyclodextrin, was investigated. Analytical methods are proposed for the determination of the three dyes in mixtures without any prior separation step, by the application of the derivative technique to the normal absorption spectra, based on the use of cetyltrimethylammonium bromide (CTAB) in 0.1 M NaOH, over the concentration range 2-50 mug ml(-1). The methods are applied for determining the three dyes in commercial food products.     

Titanocene-catalyzed cascade cyclization of epoxypolyprenes: straightforward synthesis of terpenoids by free-radical chemistry

The titanocene-catalyzed cascade cyclization of epoxypolyenes, which are easily prepared from commercially available polyprenoids, has proven to be a useful procedure for the synthesis of C(10), C(15), C(20), and C(30) terpenoids, including monocyclic, bicyclic, and tricyclic natural products. Both theoretical and experimental evidence suggests that this cyclization takes place in a nonconcerted fashion via discrete carbon-centered radicals. Nevertheless, the termination step of the process seems to be subjected to a kind of water-dependent control, which is unusual in free-radical chemistry. The catalytic cycle is based on the use of the novel combination Me(3)SiCl/2,4,6-collidine to regenerate the titanocene catalyst. In practice this procedure has several advantages: it takes place at room temperature under mild conditions compatible with different functional groups, uses inexpensive reagents, and its end step can easily be controlled to give exocyclic double bonds by simply excluding water from the medium.     

Study of the mixed-metal lanthanummagnesium-purpurin complex: spectrophotometric determination of yttrium and lanthanides

Mixtures of La(III) and Mg(II) form with purpurin (P; 1,2,4-trihydroxyanthraquinone) the mixed-metal complex LaMg₂P₅, which is extracted with methyl isobutyl ketone at pH 7.5. The molar absorptivity of the complex is 6.1 × 10⁴ l.mole⁻¹.cm⁻¹ at 570 nm and its conditional extraction constant 5 × 10¹³ l⁴.mole⁻⁴. Similar complexes have been found to be formed with yttrium, cerium, praseodymium, neodymium and samarium. The use of these complexes for the spectrophotometric determination of yttrium and lanthanides up to 15 μg has been investigated.     

Structural Diversity from the Transannular Cyclizations of Natural Germacrone and Epoxy Derivatives: A Theoretical–Experimental Study

Treatment of germacrone ( 1 ) with different electrophiles, and of its epoxy derivatives germacrone‐4,5‐epoxide ( 2 ), germacrone‐1,10‐epoxide ( 3 ) and isogermacrone‐4,5‐epoxide ( 4 ) with Brönsted/Lewis acids and Ti^III, gives rise to a great structural diversity. Thus, by using a maximum of two steps, the production of more than 40 compounds corresponding to 14 skeletons is described. Computational calculations rationalizing the structural divergence produced are also described. Finally, since some of the compounds generated are bioactive natural sesquiterpenes, the mechanisms of formation of these substances will provide new insights in their biosynthesis.     

On the reaction of iron oxides and oxyhydroxides with tannic and phosphoric acid and their mixtures

The actions of tannic acid, phosphoric acid and their mixture on lepidocrocite, goethite, superparamagnetic goethite, akaganeite, magnetite, hematite and maghemite for 1 day and 1 month were explored. It was found that these acids form iron tannates and phosphates. Lepidocrocite and magnetite were the iron phases more easily transformed with the mixture of the acids after 1 month of reaction, whereas hematite was the most resistant phase. In the case of goethite, our results suggest that in order to understand properly the action of these acids, we have to take into account its stoichiometry, surface area and degree of crystallinity.     

Electrically forced coaxial nanojets for one-step hollow nanofiber design

The outer liquid of a two-liquid coaxial electrified jet is gelled before the onset of natural instabilities to yield hollow nanofibers. By using sol-gel chemistry, innocuous solvents such as glycerol and olive oil, and electrohydrodynamics, it is possible to make such structures in a rather straightforward manner.