CCP - lect.06 - Numerical Integration

Given a function f (x) explicitly or defined at a set of n + 1 distinct tabular points, we discuss methods to obtain the approximate value of the rth order derivative f (r) (x), r ≥ 1, at a tabular or a non-tabular point and to evaluate w x a b () z f (x) dx, where w(x) > 0 is the weight function and a and / or b may be finite or infinite. 4.2 NUMERICAL DIFFERENTIATION Numerical differentiation methods can be obtained by using any one of the following three techniques : (i) methods based on interpolation, (ii) methods based on finite differences, (iii) methods based on undetermined coefficients. Methods Based on Interpolation Given the value of f (x) at a set of n + 1 distinct tabular points x 0 , x 1 , ..., x n , we first write the interpolating polynomial P n (x) and then differentiate P n (x), r times, 1 ≤ r ≤ n, to obtain P n r () (x). The value of P n r () (x) at the point x*, which may be a tabular point or a non-tabular point gives the approximate value of f (r) (x) at the point x = x*. If we use the Lagrange interpolating polynomial P n (x) = l x f x i i i n () () = ∑ 0 (4.1) having the error term E n (x) = f (x) – P n (x) = () () ... () ()! x x x x x x n n − − − + 0 1 1 f (n+1) (ξ) (4.2) we obtain f (r) (x *) ≈ P x n r () () * , 1 ≤ r ≤ n and E x n r () () * = f (r) (x *) – P x n r () () * (4.3) is the error of differentiation. The error term (4.3) can be obtained by using the formula 212

2000

Let k be a positive integer. The limiting distribution of the nonparametric maximum likelihood estimator of a k!monotone density is given in terms of a smooth stochastic process Hk described as follows: (i) Hk is everywhere above (or below) Yk, the k !1 fold integral of two-sided standard Brownian motion plus (k!/(2k)!)t2k when k is even (or odd). (ii) H

2004

Abstract Let k be a positive integer. The limiting distribution of the nonparametric maximum likelihood estimator of ak− monotone density is given in terms of a smooth stochastic process Hk described as follows:(i) Hk is everywhere above (or below) Yk, the k− 1 fold integral of two-sided standard Brownian motion plus (k!/(2k)!) t2k when k is even (or odd).(ii) H(2k− 2) k is convex.(iii) Hk touches Yk at exactly those points where H (2k− 2) k has changes of slope.

In this paper we present with proof an explicit (n+1)-points method to approximate the m-th partial derivative to functions of two variables included mixed derivatives, then we generalize this method to functions of k variables, hence we can derive (r,m)-points …nite di¤erence formulas to functions of two variables and N-points …nite di¤erence formulas to general functions of k variables, our work is generalized for the …nite di¤erence formula which applicable for equally and unequally spaced data.

Bulletin of the Brazilian Mathematical Society, New Series, 2005

A near-identity nilpotent pseudogroup of order m ≥ 1 is a family f 1 , . . . , f n : (−1, 1) → R of C 2 functions for which: |f i − id| C 1 < for some small positive real number < 1/10 m+1 and commutators of the functions f i of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.

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2005

Due to its intimate relation to Spectral Theory and Schr\"{o}dinger operators, the multivariate moment problem has been a subject of many researches, so far without essential success (if one compares with the one--dimensional case). In the present paper we reconsider a basic axiom of the standard approach - the positivity of the measure. We introduce the so--called pseudopositive measures instead. One of our main achievements is the solution of the moment problem in the class of the pseudopositive measures. A measure \ $\mu$ is called pseudopositive if its Laplace-Fourier coefficients $\mu_{k,l}(r) ,$ $r\geq0,$ in the expansion in spherical harmonics are non--negative. Another main profit of our approach is that for pseudopositive measures we may develop efficient ''cubature formulas'' by generalizing the classical procedure of Gauss--Jacobi: for every integer \ $p\geq1$ we construct a new pseudopositive measure $\nu_{p}$ having ''minimal support'' and such that $\mu(h) =\nu_{p}(h) $ for every polynomial $h$ with $\Delta^{2p}h=0.$ The proof of this result requires application of the famous theory of Chebyshev, Markov, Stieltjes, Krein for extremal properties of the Gauss-Jacobi measure, by employing the classical orthogonal polynomials $p_{k,l;j},$ $j\geq0,$ with respect to every measure $\mu_{k,l}.$ As a byproduct we obtain a notion of multivariate orthogonality defined by the polynomials $p_{k,l;j}$. A major motivation for our investigation has been the further development of new models for the multivariate Schr\"{o}dinger operators, which generalize the classical result of M. Stone saying that the one--dimensional orthogonal polynomials represent a model for the self--adjoint operators with simple spectrum.