Sequences of nonlinear differential equations with related solutions

Summary A second-order nonlinear differential equation which occurs (together with variants of it) in many problems of applied mathematics, physics and engineering is here reduced to a first-order equation. This equation contains a parameter which is a quadratic rational function of two parameters appearing in the original equation. By applying a certain identity for a quadratic rational function, two (finite or infinite) sequences of nonlinear differential equations are generated whose solutions are determinable whenever the solution of any equation belonging to a sequence is known. The cases amenable to exact solution by quadrature are given. Entrata in Redazione il 16 luglio 1968.     

A new domino Knoevenagel-hetero-Diels-Alder reaction: an efficient catalyst-free synthesis of novel thiochromone-annulated thiopyranocoumarin derivatives in aqueous medium

An efficient catalyst-free synthesis of novel pentacyclic thiochromone-annulated thiopyranocoumarin derivatives is achieved via domino Knoevenagel-hetero-Diels-Alder reaction of 4-hydroxy dithiocoumarin and O-acrylated salicylaldehyde derivatives in H₂O as solvent. The products are formed in good yields with high regio- and stereo-selectivity.     

Synthesis,characterization and single crystal structure determination of aluminum alkoxydisilanolates: precursors for silica–alumina composite

Several novel aluminum alkoxydisilanolate complexes were prepared by reaction of triphenylsilanol with aluminum 2‐methoxyethoxide, aluminum 2‐ethoxyethoxide, aluminum sec‐butoxide and aluminum iso‐propoxide. All new complexes, [(Ph₃SiO)₂Al(OR)]₂ [where R = CH₂CH₂OCH₃ (1), CH₂CH₂OC₂H₅ (2), CH(CH₃)CH₂CH₃ (3) and CH(CH₃)₂ (4)] were characterized by elemental analysis, mass spectrometry and infrared spectroscopy (IR), as well as ¹H, ¹³C, ²⁹Si and ²⁷Al NMR spectroscopies. The solid‐state structures of the representative compound 2 and 4 were also verified by single‐crystal X‐ray analyses. Complexes 2 and 4 are dimers having distorted trigonal bipyramidal and tetrahedral coordination at the aluminum center, respectively. The ²⁷Al NMR spectrum of compound 2 showed that the solid‐state structure of the complex was not retained in solution, and tetracoordinated aluminum was found in solution in contrast to the pentacoordinated geometry in the solid state. The hydrothermal treatment of 1 and 4 at 200 °C and the subsequent calcination at 1000 °C resulted in the formation of alumina–silica composite (4SiO₂·Al₂O₃) with γ‐alumina in the silica matrix. Copyright © 2010 John Wiley & Sons, Ltd.     

Large-eddy simulation of transitional flows using a co-located grid

A large-eddy simulation (LES) of a transitional separated flow over a plate with a semi-circular leading at low (<0.2%) and high (5.6%) free-stream turbulence (FST) has been performed, using a co-located grid with the Rhie–Chow pressure smoothing. A numerical trip is used to produce a high FST level and a dynamic subgrid-scale model is also employed in the current study. The entire transition process leading to breakdown to turbulence has been shown clearly by the flow visualisations using instantaneous spanwise vorticities, and the differences between the low- and high-FST cases are clearly visible. Coherent structures are also visualised using isosurfaces of the Q-criterion, and for the high-FST case, the spanwise-oriented quasi-two-dimensional rolls, which are clearly present in the low-FST case, are not visible anymore. Detailed quantitative comparisons between the present LES results and experimental data and the previous LES results at low FST using a staggered grid have been done and a good agreement has been obtained, indicating that the current LES using a co-located grid with pressure smoothing can also predict transitional flows accurately.     

Silica‐supported terpyridine palladium(II) complex as an efficient and reusable catalyst for Heck and Suzuki cross‐coupling reactions

Silica‐supported terpyridine palladium(II) was prepared and used as an effective and recyclable catalyst in Mizoroki–Heck and Suzuki–Miyaura coupling reactions. The catalyst was very effective for the Mizoroki–Heck reaction of aryl halides with olefins and conversion was in most cases excellent. The catalyst showed good thermal stability (up to 230 °C) and could be recovered and reused for four reaction cycles. The Suzuki coupling of aryl iodides with aryl boronic acids in the presence of the catalyst was also investigated and the reaction proceeded with a short reaction time and excellent conversion. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel one-pot domino reaction for the synthesis of 2-acetyl-3-(phenylamino)indolizine-1-carboxamide derivatives

2-Acetyl-3-(phenylamino)indolizine-1-carboxamide derivatives were obtained via a one-pot domino reaction of alkyl or aryl isocyanides and pyridine-2-carbaldehyde in the presence of acetoacetanilide in toluene without any prior activation or modification.     

Kinetic and Mechanistic Insights into the Pathway Leading to Cyclic Crystalline Phosphorus Ylide Formation in the Presence of 3‐Chloropentane‐2,4‐dione: Theoretical and Stopped‐Flow Approaches

In this work, three speculative mechanisms of the reaction between triphenylphosphine and dimethyl acetylendicarboxylate in the presence of 3‐chloropentane‐2,4‐dione were energetically and thermodynamically developed using quantum mechanical calculations and were profoundly compared with stopped‐flow and UV spectrophotometry approaches. The third speculative mechanism that led to the five‐membered ring structure was experimentally and theoretically favorable. The five‐membered ring structure of product was characterized by X‐ray crystallographic data. Also, steps 1 and 2 of the third mechanism were determined as fast and rate‐determining steps, respectively. The experimental kinetic evidence of the formation and decay of intermediate in steps 1 and 2 (fast and rate‐determining steps, respectively) was compatible with theoretical data. Experimental kinetic data were recognized for overall reaction along with activation parameters for fast and rate‐determining steps of the reaction. Theoretical kinetic data (k and E_a) and activation parameters (ΔG^≠, ΔS,^≠ and ΔH^≠) were calculated for each step and overall reactions.     

Kinetic Investigation of the Reaction between Triphenylphosphine,Dialkyl Acetylenedicarboxylate,and Carbazole by the UV Spectrophotometry Technique

The major objectives of the work undertaken were to carry out kinetic studies of the reaction between triphenylphosphine and dialkyl acetylenedicarboxylate in the presence of strong NH-acids, such as carbazole. To determine the kinetic parameters of the reaction, it was monitored by the UV spectrophotometer technique. The values of the second order rate constant (k ₂ ) were automatically calculated using standard equations within the program when the second order fits of the mentioned reactions were automatically drawn by the software associated with a Cary UV spectrophotometer model Bio-300 at an appropriate wavelength. At the temperature range studied, the dependence of the second order rate constant (Ln k) on reciprocal temperature was in agreement with the Arrhenius equation. This provided the relevant plots to calculate the activation energy of all reactions. Furthermore, useful information was obtained from studies of the effect of solvent and different alkyl groups within the dialkyl acetylenedicarboxylates on the rate of reactions.