The purpose of this paper is two-fold. Firstly, we explore some of the intimate connections between discrete least squares processes and quadratures. Secondly, we present an algorithm to construct Gauss-type integration formulas, and consider briefly the method proposed by Gautschi [2].

We study the construction of a quadrature rule which allows the simultaneous integration of a given function with respect to different weights. This construction is built on the basis of simultaneous Padé approximation of a Nikishin system of functions. The properties of these approximants are used in the proof of convergence of the quadratures and positivity of the corresponding quadrature coe...

We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neug...

Ermakov systems possessing Noether point symmetry are identified among the Ermakov systems that derive from a Lagrangian formalism and, the Ermakov invariant is shown to result from an associated symmetry of dynamical character. The Ermakov invariant and the associated Noether invariant, are sufficient to reduce these systems to quadratures. PACS number(s): 02.30.Hg, 02.90.+p, 03.20.+i