Approximate solutions of partial differential equations by some Meshfree Greedy Algorithms

In this article, we use some greedy algorithms to avoid the ill‐conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well‐known PDE examples are solved and compared with the full unsymmetric Kansa's technique. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1884–1899, 2017     

Synthesis of novel spiro-fused pyrazolo[4′,3′:5,6]pyrido[2,3-d]pyrimidines

New pyrazolo[4′,3′:5,6]pyrido[2,3-d]pyrimidines, with an associated spiro-3,3′-oxindole attachment, were prepared by three-component combinations of 5,6-dihydro-4H-pyrrolo[3,2,1-ij]quinoline-1,2-dione with a pair of reactants chosen from a pyrazol-5-one, a pyrazole-5-amine, a barbituric acid, or a 6-aminouracil.     

On dimension of the Schur multiplier of nilpotent Lie algebras

Let L be an n-dimensional non-abelian nilpotent Lie algebra and $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.     

Quadratic electro-optic effect and electro-absorption process in CdSe-ZnS-CdSe structure

The third order nonlinearity susceptibility is obtained as a function of wavelength for CdSe-ZnS-CdSe structure. Numerical calculations show that the quadratic electro-optic and electro absorption effects of this structure depend on the parameters such as structure size, relaxation time and pump photon energy. By changing shells thickness, these structures can be designed with the susceptibility peak from the visible to infrared regions of the spectrum. The intensity and position of the third order nonlinearity susceptibility peaks depend on outer shell thickness and radius of the core.     

Ferric Perchlorate: An Efficient Reagent for Regio- and Stereoselective Alcoholysis and Hydrolysis of Epoxides

Regio- and stereoselective ring opening reaction of epoxides is achieved efficiently by aliphatic alcohols and water in the presence of catalytic amounts of ferric perchlorate.     

Crystallisation behaviour during tensile loading of laser treated Fe–Si–B metallic glass

Effect of tensile loading on crystallisation behaviour of as-cast and laser thermal treated Fe–Si–B metallic glass foils was investigated. Tensile loading lacked any marked influence on the crystallisation behaviour of as-cast and structurally relaxed laser-treated metallic glass foils. Furthermore, the average crystallite/grain size in partially crystallised laser-treated metallic glass foil was nearly equal to the average crystallite/grain size in the region away from the fracture of the same partially crystallised laser-treated metallic glass foil after tensile loading. However, a significant crystallite/grain growth/coarsening of the order of two and half times was observed in the fractured region compared to the region around it for the laser-treated partially crystallised metallic glass foils. The simultaneous effects of stress generation and temperature rise during tensile loading were considered to play a key role in crystallite/grain growth/coarsening.     

Efficient 2,4,6‐trichloro‐1,3,5‐triazine‐catalyzed synthesis of 2‐arylbenzothiazoles and bisbenzothiazoles by condensation of 2‐aminithiophenol with aldehydes under mild conditions

2,4,6‐Trichloro‐1,3,5‐triazine efficiently catalyzed the condensation reactions between 2‐aminothiophenol and aromatic aldehydes to afford 2‐arylbenzothiazolles in good‐to‐excellent yields. Simple and mild reaction conditions, the use of a cheap catalyst and easy work up, and isolation are notable features of this method. J. Heterocyclic Chem., (2011).