An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Energy Conservation for a Scalar Field in Linear Gravity

In this paper we derive a Gaussian flux integral for a metric in the weak field limit. We shall prove that the energy integral for an approximately linear metric due to a scalar field, will possess a same form for the Newtonian mass.     

Synthesis of chromene derivatives in the presence of mordenite zeolite/MIL‐101 (Cr) metal–organic framework composite as catalyst

In the present study, the synthesis of mordenite zeolite/MIL‐101(Cr) metal–organic framework (MOF) composite [MOR/MIL‐101(Cr)] using the ship in a bottle method was suggested. The properties of prepared composite and individual MOF and MOR zeolite were characterized by X‐ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), nitrogen adsorption–desorption measurement, and thermogravimetric analysis (TGA). The XRD results indicated diffraction peaks for each compound (MOR and MOF) in composite. The SEM and TEM images showed the formation of plates MOR (with size of 2.5 × 3 μm) along with spherical particles MIL‐101. The Brunauer–Emmett–Teller results showed that the surface area of the composite was smaller than individual MOF and MOR zeolite. Based on TGA plots, the hybrid zeolite/MOF composite was more thermally stable compared with the isolated MIL‐101(Cr). The composite was functionalized by post‐synthetic modification to obtain acid–base bifunctionality (H‐MOR/MIL‐101‐ED) for the synthesis of chromene derivatives. The acidity from framework Al‐O(H)‐Si sites in MOR and basicity from amine groups in MIL‐101 were obtained by post‐synthetic modification.     

Synthesis,Crystal Structure,and Characterization of Na7Cu4(AsO4)5

Abstract

Sodium copper (II) arsenate Na₇Cu₄(AsO₄)₅ has been grown by conventional high-temperature, solid-state methods in molten-salt media. It was characterized by single crystal X-ray diffraction (XRD), thermal analysis (DTA–TGA), scanning electron microscopy (SEM), semiquantitative energy dispersive spectroscopy analysis (EDS), and vibrational spectroscopy. Na₇Cu₄(AsO₄)₅ exhibits a three-dimensional framework built up of CuO₅, CuO₄, and AsO₄ polyhedra, with intersecting channels in which the Na⁺ cations are located. The three-dimensional cohesion of the framework results from Cu–O–As bridges. CuO₅ and CuO₄ polyhedra are elongated due to the Jahn–Teller effect, whereas AsO₄ tetrahedra are almost regular. This new structural model is validated by the charge distribution (CD) analysis. The infrared and Raman spectra confirmed the presence of AsO₄ tetrahedra.

[Supplementary materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfer, and Silicon and the Related Elements for the following free supplemental files: Additional tables and figures.]     

Many-body and quantum effects in the surface tension and surface energy of liquid neon and argon using the Fowler’s approximation

The Fowler’s expression for calculation of the reduced surface tension and surface energy has been used with Lennard-Jones (LJ) and two-body Hartree-Fock dispersion (HFD)-like potentials for neon and argon, respectively. The required radial distribution functions (RDFs) have been used from two recently determined expressions in the literature and a new equation proposed in this work. Quantum corrections for neon system have been considered using the Feynman-Hibbs (FH) and Wigner-Kirkwood (WK) approaches. To take many-body forces into account for argon system, the simple three-body potentials of Wang and Sadus (2006) [33] and Hauschild and Prausnitz (1993) [30] used with the HFD-like potential without requiring an expensive three-body calculation. The results show that the quantum and three-body effects improve the prediction of the surface tension of liquid neon and argon using the Fowler’s expression.     

Synthetic Studies Towards Taxol Analogs: Chemoselective Cleavage of C-5 Cinnamoyl Group in Taxane Group of Diterpenoids with Hydroxylamine

Taxane group of diterpenoids 2 and 3 which possess cinnamoyl moiety at C-5 position underwent selective cleavage to C-5 hydroxy compounds 4 and 5 on treatment with hydroxylamine. The resulting compounds were characterised based on their spectral data.     

Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field

In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.