New labdane diterpenes from the stem bark of Larix laricina

Two new labdane diterpenes, cis-19-hydroxyabienol (1) and 8alpha-hydroxy-12Z,14-labdadien-19-al (2), along with another labdane described for the first time in the genus Larix, 19-acetoxy-13S-hydroxy-8(17),14-labdadiene (3) and a stilbene, 3-methoxy-3,3',5'-trihydroxystilbene (4), were isolated from the stem bark of Larch (Larix laricina). Their structures were established by standard chemical and spectroscopic methods. Compounds 2 and 3 were shown to be moderately cytotoxic against A-549, DLD-1 and normal skin fibroblast cell lines, WS1. Compound 1 was found to be selectively active against colon carcinoma cell lines, DLD-1.     

Rational differential operators and their kernels

A linear differential/integral operator is associated to a rational matrix in a natural way, and the behavior is defined to be the kernel of this operator. Conditions under which two rational matrices have the same behavior are derived.     

Positive kernel operators in L^{p(x)} spaces

A characterization of a weight $v$ governing the boundedness/compactness of the weighted kernel operator $K_v$ in variable exponent Lebesgue spaces $L^{p(\cdot )}$ is established under the log-Hölder continuity condition on exponents of spaces. The kernel operator involves, for example, weighted variable parameter fractional integral operators. The distance between $K_v$ and the class of compact integral operators acting from $L^{p(\cdot )}$ to $L^{q(\cdot )}$ (measure of non-compactness) is also estimated from above and below.     

How to define the dual of a higher-dimensional linear system

The classical duality of linear systems is generalized to higher dimensions. Consequently, a new insight is given to the concepts of controllability and observability, and autonomy.     

Bio-Guided Fractionation of Retama raetam (Forssk.) Webb & Berthel Polar Extracts

The fractionation of the methanolic extract (MeOH-E) of Retama raetam (Forssk.) Webb & Berthel and further analysis by thin layer chromatography resulted in four fractions (F1, F2, F3 and F4) that, in parallel with the MeOH-E, were screened for antioxidant, cytotoxic, antidiabetic and antibacterial properties. In addition, chemical characterization of their bioactive molecules was performed using LC-DAD-ESI/MSⁿ. The results indicated that F3 was the most promising regarding antioxidant and cytotoxicity abilities, possibly due to its richness in flavonoids class, particularly isoflavones. In turn, F1 was characterized by the presence of the most polar compounds from MeOH-E (organic acids and piscidic acid) and showed promising abilities to inhibit α-amylase, while F4, which contained prenylated flavonoids and furanoflavonoids, was the most active against the tested bacteria. The gathered results emphasize the distinct biological potentials of purified fractions of Retama raetam.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.