Ground State of the Conformal Flow on 𝕊3

We consider the conformal flow model derived in Bizoń et al. (2017) as a normal form for the conformally invariant cubic wave equation on 𝕊³. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions that we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting in their own right due to a supersymmetric structure), we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy. © 2019 Wiley Periodicals, Inc.     

Antioxidative properties of <Emphasis Type="Italic">Sambucus nigra</Emphasis> extracts

Potential protective effects of elderberry (Sambucus nigra) extracts against oxidative degradation of hyaluronan (HA) were detected in vitro. To induce free-radical-mediated HA degradation, Weissberger’s biogenic oxidative system, which mimics the situation of acute inflammation, was applied. Time- and dose-dependent changes of dynamic viscosity of the HA solutions in the presence and absence of two elderberry extracts produced in 2006 and 2012 were recorded by rotational viscometry (RV). Radical scavenging capacity of both extracts was investigated by the spectrocolorimetric ABTS [2,2′-azinobis(3-ethylbenzothiazoline-6-sulfonic acid) diammonium salt] assay and the “inverted” ABTS assay. Oxygen consumption of the system oxidizing HA either in the absence or presence of the elderberry extracts was determined. The results of RV revealed that an addition of the newer extract (2012) promoted the inhibition of HA degradation more markedly compared to the older extract (2006). The same effect of both extracts on ABTS^·+ scavenging was observed. Inverted ABTS assay demonstrated that colorful substances were not responsible for the radical-scavenging activity. Results of oximetry demonstrated that the effect of the extract from 2006 was more significant than that of the extract from 2012.     

Influence of Extraction Solvent on the Phenolic Profile and Bioactivity of Two Achillea Species

The phenolic composition, as well as the antioxidant and antimicrobial activities of two poorly investigated Achillea species, Achillea lingulata Waldst. and the endemic Achillea abrotanoides Vis., were studied. To obtain a more detailed phytochemical profile, four solvents with different polarities were used for the preparation of the plant extracts whose phenolic composition was analyzed using UHPLC-MS/MS (ultra-high performance liquid chromatography-tandem mass spectrometry). The results indicate that both of the investigated Achillea species are very rich in both phenolic acids and flavonoids, but that their profiles differ significantly. Chloroform extracts from both species had the highest yields and were the most chemically versatile. The majority of the examined extracts showed antimicrobial activity, while ethanolic extracts from both species were potent against all tested microorganisms. Furthermore, the antioxidant activity of the extracts was evaluated. It was found that the ethanolic extracts possessed the strongest antioxidant activities, although these extracts did not contain the highest amounts of detected phenolic compounds. In addition, several representatives of phenolic compounds were also assayed for these biological activities. Results suggest that ethanol is a sufficient solvent for the isolation of biologically active compounds from both Achillea species. Moreover, it was shown that the flavonoids naringenin and morin are mainly responsible for these antimicrobial activities, while caffeic, salicylic, chlorogenic, p-coumaric, p-hydroxybenzoic, and rosmarinic acid are responsible for the antioxidant activities of the Achillea extracts.     

Delamination of sodium montmorillonite in PA6 in the presence of melamine polyphosphate

This study describes the influence of melamine polyphosphate (MPP) flame retardant addition on layered structure of virgin sodium montmorillonite (MMT) in PA6/MMT system obtained in extrusion process. It was found that the extrusion of the PA6/MMT binary system in co‐rotating twin screw extruder gives intercalated clay nanocomposites, while during extrusion of ternary PA6/MMT/MPP composite, full exfoliation of clay was obtained in the system. Structure of the composites was proved utilizing wide angle X‐ray scattering and transmission electron microscopy techniques. Exfoliation of MMT in ternary PA6/MMT/MPP system was also confirmed by rheological studies (viscosity and creep measurements) which were carried out using rotational rheometer. Young's modulus of PA6/MMT/MPP was found to be ~25% greater comparing to PA6/MPP of the same filling level. Cone calorimetry experiments proved decrease of heat release rate peak of PA6/MMT/MPP by ~55% comparing to PA6/MPP with the same total filling level. Copyright © 2017 John Wiley & Sons, Ltd.     

Optimal double control problem for a PDE model of goodwill dynamics

We propose a new optimal control model of product goodwill in a segmented market where the state variable behaviour is described by a partial differential equation of the Lotka–Sharp–McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.     

Targeted Hybrid Nanocarriers as a System Enhancing the Skin Structure

The skin is constantly exposed to external and internal factors that disturb its function. In this work, two nanosystems-levan nanoparticles and a surfactin-stabilized nanoemulsion were preserved (tested for microbial growth) and characterized (size, polydispersity, Zeta potential, and stability). The nanosystems were introduced in the model formulations-cream, tonic, and gel, and confirmed by TEM. The analysis showed that nanoemulsion has a spherical morphology and size 220–300 nm, while levan nanoparticles had irregular shapes independently of the use of matrix and with particle size (130–260 nm). Additionally, we examined the antiradical effect of levan nanoparticles and nanoemulsion in the prototype of formulations by scavenging DPPH (2,2-diphenyl-1-picrylhydrazyl; EPR spectroscopy). The model cream with both nanosystems and the whole range of products with nanosystems were evaluated in vivo for hydration, elasticity, smoothness, wrinkles and vascular lesions, discoloration, respectively. The cream improved skin condition in all tested parameters in at least 50% of volunteers. The use of more comprehensive care, additionally consisting of a tonic and gel, reduced the previously existing skin discoloration to 10.42 ± 0.58%. The presented prototype formulations are promising in improving skin conditions.