Infinitely Many Solutions of Nonlocal Kirchhoff-Type Equations via Perturbation Methods

Mathematical Notes - We study the multiplicity of weak solutions to the boundary-value problem $$\begin{alignedat}{2} - M\biggl(\iint_{\mathbb R^{2N}}|u(x)-u(y)|^2 K(x-y)\,d x\,d y\biggr)\mathscr...     

Two nontrivial solutions of boundary-value problems for semilinear Δ<Subscript><Emphasis Type="Italic">γ</Emphasis></Subscript>-differential equations

In this paper, we study the existence of multiple solutions for the boundary-value problem

$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$

where Ω is a bounded domain with smooth boundary in R ^N (N ≥ 2) and Δ _γ is the subelliptic operator of the type

$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$

We use the three critical point theorem.

     

Study of diffusion and local structure of sodium-silicate liquid: the molecular dynamic simulation

A systematic analysis on sodium-silicate melt with various silica contents was carried out. The simulation revealed two diffusion mechanisms occurred in the melt: the bond-breaking and hopping between sites. The local structure was analyzed through T-simplexes. It was revealed that T-clusters have a non-spherical shape and represent the diffusion channel, in which Na atoms are dominant, but no any O atoms are located. The SiO₂-poor melt acquires a long channel. In contrast, the SiO₂-rich melt consists of unconnected short channels. The simulation also revealed the immobile and mobile regions which differ in local structure and constituent composition. We propose a new CL-function to characterizing the spatial distribution of different atom component. The spatial distribution of mobile and immobile atoms is found quite different. In particular, the immobile atoms are concentrated in high-density regions possessing very large density of immobile atoms. The spatial distribution of mobile atoms in contrast is more homogeneous.     

Influence de la structure sur les proprietes mesomorphes des polyesters—I

A new series of thermotropic liquid crystalline polymers has been synthesized. Mesogenic elements are separated by flexible spacers —(CH₂)_n— in the backbone. The thermal properties of these polymers are analogous to liquid crystals.     

Picone’s Identity For ∆γ -Laplace Operator and its Applications

Ukrainian Mathematical Journal - We prove a nonlinear analog of Picone’s identity for the ?γ -Laplace operator. As an application, we give a Hardy-type inequality and the Sturmian...     

Anti-inflammatory constituents from Psychotria prainii H. Lév

One new and three known compounds were isolated from the ethanol extract of Psychotria prainii aerial parts. By means of spectroscopic methods, their structures were elucidated to be deacetylasperulosidic acid 6-ethyl ether (1), asperulosidic acid (2), asperuloside (3) and obtucarbamates C (4). The isolated compounds were evaluated for their inhibitory effect on NO production in LPS-stimulated RAW264.7 cells. Among them, compounds 2 and 4 exhibited strong effect with the IC₅₀ values of 5.75 ± 0.85 and 6.92 ± 0.43 μM, respectively. This is the first report for the chemical composition and biological activity of P. prainii.     

Etude structurale aux rayons-x d'un polyester nematique thermotrope

The structure of a mesomorphic polyester, studied by X-ray diffraction in the anisotropic phase, is very similar to a typical nematic. Orientation of the nematic phase by a magnetic field (0.3 T) is preserved during solidification of the polymer which then appears as a very homogeneous fibre.     

Modification of Rubber by Cardanol-Formaldehyde Resins and Epoxidized Cardanol

Abstract

Cardanol-formaldehyde resin (CF) and cardanol glycidylether (CGE) have been synthesized for reinforcing natural rubber (NR), a blend of NR and styrene-butadiene rubber (SBR), and nitrile-butadiene rubber (NBR). The novolac CF resin reinforced the NR, SBR, and NBR. The resolic CF is not only a reinforcing agent but also a hardener (crosslinking agent) for NBR by means of the methylol groups of CF with the CN- group of NBR. The CGE resin was synthesized by the epoxidation of cardanol by epichlorohydrin; it could be used as a reinforcing agent for NR and for crosslinking maleinized NR. The results of estimates of the physical properties of the vulcanisate, their DSC diagrams, and SEM showed the enhanced properties of the final products.     

Rhodamine-derived Schiff base for the selective determination of mercuric ions in water media

A new rhodamine-derived Schiff base (RS) was synthesized and its sensing property to metal ions was investigated by UV/vis and fluorescence spectroscopies. Addition of Hg2+ ions to the aqueous solution of RS gave a visual color change as well as significantly fluorescent enhancement, while other ions including Pb2+, Cd2+, Cr3+, Zn2+, Cu2+, Fe2+, Co3+, Ni2+, Ca2+, Mg2+, K+ and Na+ ions did not induce any distinct color/spectral changes, which constituted a Hg2+-selective fluorescent OFF-ON chemosensor. The Hg2+-induced ring-opening of spirolactam of rhodamine in RS resulted in the dual chromo- and fluorogenic observation.