Meshless cubature over the disk using thin-plate splines

In some recent papers, the construction of meshless interpolatory cubature formulas using radial basis functions has been studied. In particular, thin-plate splines allow us to conveniently use Green’s formula and give good results on scattered samples of small/moderate size over polygons. Here, we discuss the extension to meshless cubature over the disk and its practical implementation.     

On the generation of symmetric Lebesgue-like points in the triangle

We compute point sets on the triangle that have low Lebesgue constant, with sixfold symmetries and Gauss–Legendre–Lobatto distribution on the sides, up to interpolation degree 18. Such points have the best Lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements.     

Carbonate, acetate and phenolate phosphonium salts as catalysts in transesterification reactions for the synthesis of non-symmetric dialkyl carbonates

Methyl trioctylphosphonium methyl carbonate [P(8881)](+)[MeOCO(2)](-) was prepared by the alkylation of trioctyl phosphine with the non-toxic dimethyl carbonate. This salt was a convenient source to synthesize different ionic liquids where the methyl trioctylphosphonium cation was coupled to weakly basic anions such as bicarbonate, acetate, and phenolate. At 90-220 °C, all these compounds [P(8881)](+)X(-); X = MeOCO(2); HOCO(2); AcO; PhO were excellent organocatalysts for the transesterification of dimethyl and diethyl carbonate with primary and secondary alcohols, including benzyl alcohol, cyclopentanol, cyclohexanol, and the rather sterically hindered menthol. Conditions were optimized to operate with very low catalyst loadings up to 1 mol% and to obtain non-symmetric dialkyl carbonates (ROCO(2)R'; R = Me, Et) with selectivity up to 99% and isolated yields >90%. The catalytic performance of the investigated ionic liquids was discussed through a cooperative mechanism of simultaneous activation of both electrophilic and nucleophilic reactants.     

Cooperative nucleophilic-electrophilic organocatalysis by ionic liquids

The anionic and the cationic partners of ionic liquids may act cooperatively and independently as nucleophilic and electrophilic catalysts. This ambiphilic propensity was demonstrated by kinetically discriminating the contributions of the anion (nucleophilic catalyst) and of the cation (electrophilic catalyst) to the solvent-free Baylis-Hillman dimerization of cyclohexenone catalysed by ionic liquids.     

Triphasic liquid systems: generation and segregation of catalytically active Pd nanoparticles in an ammonium-based catalyst-philic phase

A triphasic liquid system fabricated from isooctane, aqueous base, and trioctylmethylammonium chloride/decanol promoted the formation of Pd-nanoparticles in the size range of 2-4 nm which remained immobilised in the onium phase, catalysed organic reactions, and could be recycled.     

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions for algebraic cubature.     

Quadrature-based polynomial optimization

We show that Lasserre measure-based hierarchies for polynomial optimization can be implemented by directly computing the discrete minimum at a suitable set     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

A Fast Nystr?m–Broyden Solver by Chebyshev Compression

When a Nyström–Broyden method is used to solve numerically Urysohn integral equations, the main problem is to evaluate the action of the integral operator at a low cost. Here we suitably approximate the relevant discrete integral operator of dimension n by its m-degree truncated Chebyshev series expansion (with mn), reducing the complexity from the basic O({n ²}) to O(mn). A technique to evaluate cheaply such m is presented. Several linear and nonlinear examples are considered.     

Computing Fekete and Lebesgue points: Simplex, square, disk

We have computed point sets with maximal absolute value of the Vandermonde determinant (Fekete points) or minimal Lebesgue constant (Lebesgue points) on three basic bidimensional compact sets: the simplex, the square, and the disk. Using routines of the Matlab Optimization Toolbox, we have obtained some of the best bivariate interpolation sets known so far.