The uniaxial orientational order in a macromolecular system is usually specified using the Hermans factor which equivalent to second moment of system's orientation distribution function (ODF) expanded terms Legendre polynomials. In this work, we show that for aligned materials are two-dimensional (2D) or have measurable 2D intensity distribution, such as carbon nanotube (CNT) textiles, not appr...

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then p...

Let p > 3 be a prime, and let Rp be the set of rational numbers whose denominator is not divisible by p. Let {Pn(x)} be the Legendre polynomials. In this paper we mainly show that for m,n, t ∈ Rp with m 6≡ 0 (mod p), P[ p 6 ](t) ≡ − (3 p ) p−1 ∑ x=0 (x3 − 3x + 2t p ) (mod p)

By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or scaled images using complex or geometric moments. However, the derivation of these functions is not based on Legendre polynomials. This is mainly due to the fa...

In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution of non-linear system of algebraic equations usi...

In this article, we introduce a new class of generalized Humbert-Hermite polynomials two variables. We are conscious that the present paper was inspired by unified presentation two-variable Humbert’s generalize well-known Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinnsy, Horadam-Pethe, Djordjevi’c, Gould, Milovanovi’c and Pathan, Khan polynomials. Also, results analyzing pa...

It is often the case that the exact moments of a statistic of the continuous type can be explicitly determined, while its density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. The density approximants discussed in this article are based on the first n exact moments of the corresponding distributions. A unified semiparametric approach to...

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as sl(2,C)-modules. As finite-dimensional irreducible sl(2,C)-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain ...

The three-dimensional high-order simulation algorithm HOSIM is developed to simulate complex nonlinear and non-Gaussian systems. HOSIM is an alternative to the current MP approaches and it is based upon new high-order spatial connectivity measures, termed high-order spatial cumulants. The HOSIM algorithm implements a sequential simulation process, where local conditional distributions are gener...

In this paper an iterative method based on shifted Legendre polynomials is presented to obtain the approximate solutions of optimal control problems subject to integral equations. The operational matrices of integration and product of shifted Legendre polynomials for solving integral equation is employed. The methodology is based on the parametrization of control and state functions. This conve...