A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

We prove a Liouville-type theorem for semilinear parabolic systems of the form

$$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$

in the whole space \({\mathbb R}^N\times {\mathbb R}\). Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for \(m=2\) in dimension \(N\le 2\), and partial results in higher dimensions in the range \(p< N/(N-2)\). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions \(N\ge 3\). In particular, our results solve the important case of the parabolic Gross–Pitaevskii system—i.e. the cubic case \(r=1\)—in space dimension \(N=3\), for any symmetric (m, m)-matrix \((\beta _{ij})\) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space \({\mathbb R}^N_+\times {\mathbb R}\). As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

     

Inverting the G‐Tetrad Polarity of a G‐Quadruplex by Using Xanthine and 8‐Oxoguanine

G‐quadruplexes are four‐stranded nucleic acid structures that are built from consecutively stacked guanine tetrad (G‐tetrad) assemblies. The simultaneous incorporation of two guanine base lesions, xanthine (X) and 8‐oxoguanine (O), within a single G‐tetrad of a G‐quadruplex was recently shown to lead to the formation of a stable G?G?X?O tetrad. Herein, a judicious introduction of X and O into a human telomeric G‐quadruplex‐forming sequence is shown to reverse the hydrogen‐bond polarity of the modified G‐tetrad while preserving the original folding topology. The control exerted over G‐tetrad polarity by joint X?O modification will be valuable for the design and programming of G‐quadruplex structures and their properties.     

Neutron-diffraction study of the crystal structure of BaTiO<Subscript>3</Subscript> ferroelectric doped with iron

The crystal structure of iron-doped barium titanate BaTi_1–x Fe _x O₃ is studied by neutron diffraction in the range of 0 ≤ x ≤ 0.12. At low concentrations of iron, x < 0.01, and at room temperature, these compounds have a polar structure with tetragonal symmetry with space group P4mm. The temperature of the transition of the tetragonal ferroelectric phase into the cubic paraelectric phase with space group Pm \(\bar 3\) m for an iron concentration of x = 0.01 is 390 K (for pure BaTiO₃, it is 410 K). At an iron concentration of x = 0.07, the crystal structure of the studied compounds varies, and it is described by the centrosymmetric hexagonal space group P6₃/mmc. The structural parameters of various phases of compound BaTi_1–x Fe _x O₃ are determined from the experimental data.     

A new flavan from the Drynaria bonii H. Christ rhizomes

In Vietnam, the medicinal plant Drynaria bonii H. Christ is used for the treatment of osteoporosis, bone fractures, and stimulate the growth of hair, treat tinnitus (Ho 2002; Loi 2004). In this article, experiments were designed to investigate the proliferation activity of ethanol, n-hexane, chloroform, ethyl acetate and methanol extracts from D. bonii rhizomes on MG-63 human osteoblast-like cells. The results showed that methanol and hexane extracts had the ability to proliferate MG-63 cells at the concentration varying from 0.1 to 0.01 μg/mL. In particular, at the concentration of 0.01 μg/mL, hexane and methanol extracts illustrated the highest proliferation ratio with 9.31% and 6.16%, respectively. By column chromatography, a new compound named drynaether A (1) and five known compounds uracil (2), 4′-hydroxy-7-methoxyflavan (3), kaempferol (4), indole-3-carboxylic acid (5) and protocatechuic acid (6) were isolated and identified from the methanol extract.     

A new dihydrofurocoumarin from the fruits of Pandanus tectorius Parkinson ex Du Roi

From the fruit of Pandanus tectorius Parkinson ex Du Roi, one new dihydrofurocoumarin, named pandanusin A (1) and 15 known compounds, including one furanocoumarin (2), two coumarins (3, 4), four lignans (5–8), one neolignan (9), two flavonoids (10, 11), three phenolics (12–14), one monoglyceride (15) and one monosaccharide (16) were isolated by various chromatography methods. Among them, compounds (3–5) were obtained from the Pandanus genus for the first time and compounds (9–14, 16) were reported from this species for the first time. Their structures were elucidated by HR–ESI–MS, NMR 1D and 2D experiments and comparison with previous reported data. The α-glucosidase inhibitory activity of all compounds was measured. The isolated compounds (1–12, 14) showed better α-glucosidase inhibitory activity (IC₅₀ = 42.2, 36.5, 84.7, 73.2, 40.8, 26.7, 76.5, 33.8, 68.1, 14.4, 22.1, 81.5, 43.8 μM, respectively) than the standard drug acarbose (IC₅₀ = 214.5 μM).     

Hydrogenation of natural rubber having epoxy group

Hydrogenation of epoxidized natural rubber (ENR) was performed to introduce hydroxyl group to hydrogenated natural rubber. The ENR was prepared by epoxidation of deproteinized natural rubber (DPNR) with peracetic acid in latex stage. Hydrogenation of epoxidized DPNR (EDPNR) was performed with p-toluenesulfonylhydrazide in p-xylene. The resulting product, hydrogenated EDPNR (HEDPNR), was characterized by nuclear magnetic resonance spectroscopy with various pulse sequences, i.e., two-dimensional correlation spectroscopy, two-dimensional heteronuclear correlation measurements. Carbons linking up to hydroxyl group were assigned to be quaternary and tertiary groups. The HEDPNR was proved to be a polyolefine elastomer through differential scanning calorimetry.     

Switchable-polarity solvents prepared with a single liquid component

Known liquids that can reversibly switch their polarity at atmospheric pressure are all prepared as mixtures of two liquid components; we now report a series of switchable-polarity solvents that consist, in their low-polarity form, of only a single liquid component, a secondary amine. These solvents operate in a polarity range that is significantly lower than those of previously reported switchable solvents. Application to the separation and purification of a polymer and recovery of a catalyst is described.     

Versions of Ekeland’s variational principle involving set perturbations

We consider Ekeland’s variational principle for multivalued maps. Instead of dealing with directional perturbations in a direction of the positive cone of the image space, we perturb the map under question by a convex subset of the positive cone to get stronger and more general versions. Many example are provided to highlight relations of our results to existing ones, including their advantages.     

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.