The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations

In this paper, the a posteriori error estimates of Chebyshev–Petrov–Galerkin approximations are investigated. For simplicity, we choose the Poisson equation with Dirichlet boundary conditions to discuss the a posteriori error estimators, and deduce their efficient and reliable properties. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error estimators.     

Site-selective C–H nitration of N-aryl-7-azaindoles under palladium(II) catalysis

The site-selective C–H nitration reaction of 7-azaindoles with t-butyl nitrite under palladium catalysis is described. This protocol provides an efficient method for the construction of ortho-nitrated N-aryl-7-azaindoles with excellent site-selectivity and functional group compatibility. The formed 7-azaindole derivatives can be readily transformed into 7-azaindoles containing an aniline functional group under palladium-catalyzed hydrogenation conditions.     

Phragmen–lindelöf principle for weak subsolutions and cooperative reaction–diffusion systems

One of the principal techniques for treating sustems of reaction–diffusion equations is based on a comparison method using sub and super–solutions. In practice this method is much more effective if non–smooth subsolutions are allowed. In this note we extend the analysis in [2,3] for cooperative systems and prove a comparison principle for a natural and rather general class of weak subsolutions satisfying a Phragmen–Lindelöf condition. An application is then given to a biological model in involving a pair of mutualists.     

Designed Construction of SrTiO3/SrSO4/Pt Heterojunctions with Boosted Photocatalytic H2 Evolution Activity

Efficient separation of photogenerated electron–hole pairs is a crucial factor for high-performance photocatalysts. Effective electron–hole separation and migration could be achieved by heterojunctions with suitable band structures. Herein, a porous SrTiO₃/SrSO₄ heterojunction is prepared by a sol-gel method at room temperature followed by an annealing process. XRD characterization suggests high crystallinity of the heterostructure. A well-defined interface between the two phases is confirmed by high-resolution (HR)TEM. The photocatalytic H₂ evolution productivity of the SrTiO₃/SrSO₄ heterojunction with Pt as co-catalyst reaches 396.82 μmol g⁻¹ h⁻¹, which is 16 times higher than that of SrTiO₃/Pt. The boosted photocatalytic activity of SrTiO₃/SrSO₄/Pt can be ascribed to the presence of SrSO₄, which promotes the transfer and migration of photogenerated carriers by forming the heterojunction and porous structure, which provides a large amount of active sites. This novel porous heterostructure brings new ideas for the development of high-efficiency photocatalysts for H₂ release.     

Necessary and sufficient conditions for Hölder continuity of Gaussian processes

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Calculation of the vibration–rotational transition intensities of water molecules trapped in an argon matrix: stretching O–H vibrations spectral region

The frequencies and intensities of vibration–rotational transitions of water molecules in an argon matrix were calculated for temperatures of 6 and 30 K. The rigid asymmetric top approximation was used with available literature values of the effective rotational constants in the ground and excited vibrational states. The calculations were carried out by taking into account the existence of a non-equilibrium population distribution between the rotational levels of ortho- and para-water isomers. It was assumed that the temperature relaxation of the population of rotational levels is independent of the ortho- and para-isomers. Comparison of the results of the theoretical calculations with experimental literature data shows good agreement for the majority of the rotational structure lines for symmetric and antisymmetric stretching vibrations both in the frequency values and in the values of the relative intensities.     

Liquid crystalline properties and photochromism of 4-alkyl-N-(4-alkloxysalicylidene)anilines

Eight new homologous series of 4-alkyl-N-(4-alkloxysalicylidene)anilines were synthesized. The phase transitions of these homologues were determined using differential scanning calorimetry and polarizing optical microscopy. The homologues began to exhibit liquid crystalline phases as the terminal alkyl or alkoxyl chains were lengthened. In addition, some members of these homologues series exhibited photochromism in the solid state.     

Preparation of safe water–lipid mixed electrolytes for application in ion capacitor

Aqueous electrolytes are safe, economic, and environmentally friendly. However, they have a narrow potential window. On the other hand, organic electrolytes exhibit good thermodynamic stability but are inflammable and moisture sensitive. In this study, we prepared water–PEG–lipid ternary electrolytes(TEs). To combine the advantages of water, polyethylene glycol(PEG) and propylene carbonate(PC). The nonflammable mixed electrolytes exhibited a wide potential window of about 2.8 V due to the beneficial effects of PEG and PC. Using these TEs, a lithium manganate–active carbon ion capacitor could be operated at 2.4 V with an energy density of 32 Wh/kg, based on the total active electrode material(current density of 3.3 m A/cm~2). This value was significantly higher than that achieved using an aqueous electrolyte, thereby rationalizing the higher energy density.