New control laws for stabilization of a rigid body motion using rotors system

This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced.     

基于Rodrigues参数的陀螺体受控运动

经典刚体动力学中表示刚体姿态的参数中，Euler角、Cardan角和Euler参数在工程技术中使用最为普遍．近期在航天器姿态控制问题中使用Rodrigues参数的报道也引起注意，Rodrigues参数以其表达形式简明和代数运算特点而具有独特优点．航天器姿态控制系统必须具有自适应性以适应参数的变化，建立用Rodrigues参数表达的无力矩陀螺体受控运动方程，提出基于Rodrigues参数的自适应姿态控制方案，并应用Lyapunov定理证明受控运动的渐近稳定性。     

Creation Operators for the Macdonald and Jack Polynomials

Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.     

A Formula for the Exponential of a Real Skew-Symmetric Matrix of Order 4

In this short paper the formula of the exponential matrix e ^A when A is a kew-symmetric real matrix of order 4 is derived. The formula is a generalization of the well known Rodrigues formula for skew-symmetric matrices of order 3.This revised version was published online in October 2005 with corrections to the Cover Date.     

The H q -Classical Orthogonal Polynomials

This is an expository paper that aims to give an essentially self-contained survey of a so-called q-analogue of classical polynomials from their characterizations by Hahn's property and a Rodrigues formula which allows us to construct them. Integral representations of corresponding forms are given, except in some cases where the problem remains open.     

Some properties of q-biorthogonal polynomials

Almost four decades ago, Konhauser introduced and studied a pair of biorthogonal polynomials

     

Convolution integral equations with Gegenbauer function kernel

In this paper we develop a concise and transparent approach for solving Mellin convolution equations where the convolutor is the product of an algebraic function and a Gegenbauer function. Our method is primarily based on

1. the use of fractional integral/differential operators;

2. a formula for Gegenbauer functions which is a fractional extension of the Rodrigues formula for Gegenbauer polynomials (see Theorem 3);

3. an intertwining relation concerning fractional integral/differential operators (see Theorem 1), which in the integer case reads (d/dx)²ⁿ⁺¹ = (x⁻¹ d/dx)ⁿx²ⁿ⁺¹(x⁻¹ d/dx)ⁿ⁺¹.

Thus we cover most of the known results on this type of integral equations and obtain considerable extensions. As a special illustration we present the Gegenbauer transform pair associated to the Radon transformation.     

A new approach to the theory of classical hypergeometric polynomials

In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.

     

Guido Beck,Alexandre Proca,and the Oporto Theoretical Physics Seminar

We describe the pioneering attempts made by Ruy Luís Gomes (1905–1984) and other Portuguese physicists to develop a research and teaching seminar in theoretical physics at the University of Oporto in 1942–1944 under the leadership first of the refugee Austrian theoretical physicist Guido Beck (1903–1988) and then of the Romanian-French theoretical physicist Alexandre Proca (1896–1955). These efforts failed, however, owing to lack of sustained financial support from the Portuguese government and to the political repression of the Salazar regime, which dismissed Gomes and other prominent Portuguese physicists and other scientists from their university positions.     

Some Polynomial Systems Associated with a Certain Family of Differential Operators

The authors aim at presenting several (presumably new) classes of linear, bilinear, and mixed multilateral generating functions for some general systems of polynomials which are defined by means of a certain family of differential operators. Some of the generating functions considered here are associated with the Stirling numbers of the second kind. Many (known or new) consequences and applications of the results obtained in this paper are also indicated.