Copper‐catalyzed Clauson–Kass pyrroles synthesis in aqueous media

An efficient and convenient copper‐catalyzed Clauson–Kass reaction of 2,5‐dimethoxytetrahydrofuran with amines in aqueous media has been developed, providing a wide range of N‐substituted pyrroles in good yields. It is noteworthy that the Clauson–Kass reaction of 2,5‐dimethoxytetrahydrofuran with p‐phenylenediamine or m‐phenylenediamine proceeds smoothly to afford the corresponding monopyrroles and bispyrroles with high selectivity in impressive yields. A plausible mechanism for the formation of N‐substituted pyrroles has been proposed. Copyright © 2012 John Wiley & Sons, Ltd.     

Algorithm for Optimal Triangulations in Scattered Data Representation and Implementation

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.     

Envelop solitons in dusty plasmas for warm dust

A nonlinear Schrödinger equation is obtained for the warm dusty plasmas. The modulational instability of envelop soliton is investigated in this paper. Both the temperature of the dust grains and the charge variations of dust grains affect the instability regions of the dusty plasmas. It also affect the amplitude and the width of the envelop soliton.     

Covering vertices of a graph by k disjoint cycles

Let d, k and n be three integers with k3, d4k−1 and n3k. We show that if d(x)+d(y)d for each pair of nonadjacent vertices x and y of a graph G of order n, then G contains k vertex-disjoint cycles converting at least min{d,n} vertices of G.