This thesis discusses strategies for numerically solving a two dimensional integral. The difficulties here are due to numerous integrands and regions in R2. The transformation to units as the 2-simplex and the cube can simplify many integrands. An approach to arbitrary regions is to decompose them into easy ones as triangles and rectangles. For these regions formulas that approximate the two di...

In this paper, using a new identity, we study one of the famous Newton-Cotes three-point quadraturerules. More precisely Maclaurin’s quadrature rule, for which establish error estimate methodunder constraint that first derivatives belong to class preinvex functions. We also give someapplications special means as applications. believe studied inequality and resultsobtained in article will furthe...

The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding s-orthogonal polynomials, of the same degree, are independent of s. These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev co...

We revisit the problem of extending quadrature formulas for general weight functions, and provide a generalization of Patterson’s method for the constant weight function. The method can be used to compute a nested sequence of quadrature formulas for integration with respect to any continuous probability measure on the real line with finite moments. The advantages of the method include that it w...

The spline collocation method  is employed to solve a system of linear and nonlinear Fredholm and Volterra integro-differential equations. The solutions are collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. We obtain the unique solution for linear and nonlinear system $(nN+3n)times(nN+3n)$ of integro-differential equations. This approximation reduces th...

In order to approximate the integral I( f ) = ∫ b a f (x)dx, where f is a sufficiently smooth function, models for quadrature rules are developed using a given panel of n (n ≥ 2) equally spaced points. These models arise from the undetermined coefficients method, using a Newton’s basis for polynomials. Although part of the final product is algebraically equivalent to the well known closed Newto...

The solution of an integral equation arising in a covariance factorization problem is obtained by a Newton-Raphson iteration that is almost always globally convergent. Interpretations of the iterates are given, and the result is shown to specialize to known algorithms when the covariance is stationary with a rational Fourier transform.

We express the cohomology of the basic Newton stratum of some unitary Shimura varieties associated to division algebras in terms of automorphic representations of the group in the Shimura datum. 2010 Mathematics Subject Classification: Primary 11G18 ; Secondary 11F70, 11F72.

Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n ≥ 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by OL of signature (1, n−1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.