Controllability on classical Lie groups

We consider the problem of accessibility and controllability of certain bilinear systems. These evolve on Lie groups whose Lie algebras are the normal real forms of complex simple Lie algebras. Previous results by other authors were obtained under the assumption that the controlled vector field is strongly regular. Our paper is aimed at weakening this requirement, and involves relating the root structure of elements in a Lie algebra as above to the nodal connection graphs obtained from their standard matrix representations. This is in turn related to a standard irreducibility assumption on the uncontrolled vector field. The abstract results on generation of Lie algebras are of some independent interest. The work of this author was supported in part by a Fulbright grant, while she was visiting the Arizona State University, and by the Centro de Matemática da Universidade de Coimbra/INIC. The work of this author was partially supported by AFOSR Contract No. 85-0224A.     

多连通李群覆盖学习算法在图像分类上的应用

李群机器学习作为一种新的学习范式已被学术界广泛关注。根据李群的连通性质,将具有不同类别特征的研究对象映射到多连通李群空间,并从各个单连通李群空间上连线的同伦等价出发,运用覆盖的思想寻找对应不同类别的最优道路等价表示,从而用多连通李群的多值表示来呈现图像的类别信息,因此提出了多连通李群覆盖学习算法。在MPEG7_CE-Shape01_Part_B图像库的图像和MNIST手写体数字图像上进行了实验验证,结果表明与两种基于李群均值的学习算法相比,多连通李群覆盖学习算法具有较好的分类效果。     

李群均值学习算法

首先分析李群均值的计算方法,在此基础上,进一步提出李群均值学习算法,其思想是在李群流形上寻找一个由总体样本内均值的李代数元素决定的单参数子群,这个单参数子群是原李群上的一条测地线,定义样本到测地线投影的概念,同时将李群样本向该测地线投影,并尽可能使投影后各类别间的散度与类内散度比值最大化,从而实现非线性李群空间的类别判别。实验表明,基于李群均值的学习算法和KNN、FLDA算法相比,具有较好的分类效果。     

Isochronous sets of invariant control systems

Let G be a connected Lie group with Lie algebra g and Σ=(G,D) a controllable invariant control system. A subset A⊂G is said to be isochronous if there exists a uniform time T_A>0 such that any two arbitrary elements in A can be connected by a positive orbit of Σ at exact time T_A. In this paper, we search for classes of Lie groups G such that any Σ has the following property: there exists an increasing sequence of open neighborhoods (V_n)_n≥0 of the identity in G such that the group can be decomposed in isochronous rings W_n=V_n+1−V_n. We characterize this property in algebraic terms and we show that three classes of Lie groups satisfy this property: completely solvable simply connected Lie groups, semisimple Lie groups and reductive Lie groups.     

Hamiltonian point of view of non-Euclidean geometry and elliptic functions

This paper offers a new way of looking at the classical geometries and the theory of elliptic functions through Hamiltonian systems on Lie groups. In particular, the paper shows that: (i) the classical models of non-Euclidean geometries are canonically induced by bi-invariant sub-Riemannian metrices on Lie groups which act by left-actions on the underlying space; (ii) there is a class of canonical variational problems on Lie groups G whose projections on homogeneous spaces G/K generalize Euler's elasticae and include all curves of constant curvature and all -functions of Weierstrass; (iii) complex Lie groups unify non-Euclidean geometries and complex elasticae offer a distinctive look at the elliptic functions.     

李群机器学习十年研究进展

该文主要从3个方面介绍李群机器学习近年来的研究进展。首先,该文将解释为什么采用李群结构进行数据或特征描述,以此阐明李群机器学习与传统机器学习方法的区别,并且通过李群在人工智能领域的广泛应用来说明李群表示的普遍性。其次,该文概述了李群机器学习自提出以来的主要学习算法,着重强调最近的一些研究进展。最后,针对目前的研究现状,该文给出李群机器学习未来的一些研究方向。     

双线性系统可控性综述

双线性系统是一类特殊的非线性系统,广泛存在于现实世界中,如工程、经济、生物、生态等领域,被认为是最接近于线性系统的非线性系统．对双线性系统的研究已历经了近半个世纪. 作为系统最基本的属性,双线性系统可控性的研究一直以来是热点和难点.本文分别对连续双线性系统可控性和离散双线性系统可控性进行讨论, 综述了双线性系统可控性的研究.特别地,报告了近来对离散双线性系统可控性研究的新成果.最后,例举了一些可控的双线性系统例子.     

Theorems on the structure of finite dimensional estimation algebras

In this paper we proved several theorems concerning the structure of finite dimensional estimation algebras. In particular, under proper technical assumptions, we proved the following: (1) The observation of a filtering system must be linear if the estimation algebra is finite dimensional. (2) All elements of a finite dimensional estimation algebra belong to a special class of polynomial differential operators. (3) All finite dimensional estimation algebras are solvable.     

Computing the fine structure of real reductive symmetric spaces

Much of the structure of Lie groups has been implemented in several computer algebra packages, including LiE, GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces is very similar to that of the underlying Lie group and a computer algebra package for computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics. Until recently only very few algorithms existed for computations in symmetric spaces due to the fact that their structure is much more complicated than that of the underlying group.     

基于李代数的变形目标跟踪

动态几何变形是图像跟踪技术面临的突出难题之一. 本文提出基于李代数的变形目标跟踪方法, 用Gabor特征表征目标, 以仿射李群建立目标几何变形, 利用李代数和李群之间的指数映射将参数的最优化求解从欧氏空间转至光滑流形, 实现了对变形目标的稳定跟踪.从物理层面分析了目标跟踪过程中的参数几何变换的实质, 从理论上对在光滑流形上进行迭代求解的优点进行了详细分析, 并对其收敛性做出了证明.图像序列跟踪测试的对比实验表明, 本文方法较现有基于欧氏空间的算法在收敛速度、跟踪稳定性和精确性方面有显著提高.     

Metrics for 3D Rotations: Comparison and Analysis

3D rotations arise in many computer vision, computer graphics, and robotics problems and evaluation of the distance between two 3D rotations is often an essential task. This paper presents a detailed analysis of six functions for measuring distance between 3D rotations that have been proposed in the literature. Based on the well-developed theory behind 3D rotations, we demonstrate that five of them are bi-invariant metrics on SO(3) but that only four of them are boundedly equivalent to each other. We conclude that it is both spatially and computationally more efficient to use quaternions for 3D rotations. Lastly, by treating the two rotations as a true and an estimated rotation matrix, we illustrate the geometry associated with iso-error measures.

Higher order geodesics in Lie groups

For all n > 2, we study nth order generalisations of Riemannian cubics, which are second-order variational curves used for interpolation in semi-Riemannian manifolds M. After finding two scalar constants of motion, one for all M, the other when M is locally symmetric, we take M to be a Lie group G with bi-invariant semi-Riemannian metric. The Euler–Lagrange equation is reduced to a system consisting of a linking equation and an equation in the Lie algebra. A Lax pair form of the second equation is found, as is an additional vector constant of motion, and a duality theory, based on the invariance of the Euler–Lagrange equation under group inversion, is developed. When G is semisimple, these results allow the linking equation to be solved by quadrature using methods of two recent papers; the solution is presented in the case of the rotation group SO(3), which is important in rigid body motion planning.     

动载荷反演分析技术研究综述

介绍了动载荷反演分析技术工程应用背景,综述了动载荷反演分析理论研究的最新进展.从动载荷反演分析理论、工程应用等方面回顾了动载荷反演分析技术的发展历程,探讨了动载荷反演分析研究中存在的关键科学问题,对动载荷反演分析技术应用于实际工程,后续需要进一步开展的研究方向进行了展望.     

分析动力学三个问题的研究进展

分析力学的发展涉及理论的和应用的诸多方面.本文在分析力学与数学交缘的三个问题上综述分析力学的近代发展.第一是利用Lie群和Lie代数的一些成果来研究分析动力学方程的积分问题.第二是将分析力学的经典和近代积分方法应用于一般微分方程的积分问题.第三是将分析动力学方程在一定条件下化成梯度系统的方程,再用梯度系统的性质来研究力学系统的动力学行为.     

A Lie group formulation of Kane’s equations for multibody systems

We propose a Lie group approach to formulate the Kane’s equations of motion for multibody systems. This approach regards the set of rigid body transformations as the special Euclidean group SE(3). By expressing rigid body displacements as exponential maps generated from the Lie algebra se(3), it subsequently manipulates rigid body kinematics as convenient matrix operations. With this approach, all the individual quantities involved in Kane’s equations can be computed explicitly in an intrinsic manner, and the motion equations can be obtained systematically and efficiently. An example is presented to illustrate its use and effectiveness.     

广义Burgers方程的对称分类及其约化

利用李群方法对广义Burgers方程ut+f(x,t)(ux-uxx)=0的对称分类及其约化作具体讨论,其中f是关于自变量x,u的光滑函数,得到了f(x,t)的八种分类对称及相应的约化方程.该结果对于广义Burgers方程精确解的研究有重要意义.     

真实的谎言——图形中的矛盾空间问题

优秀的设计来源于意念的创造,矛盾空间是艺术空间,是创造性思维的重要表现形式之一。本文由一部影片开始,引发了对于视觉思维、矛盾空间、数学与悖论之间的关系的思考。     

On a new method for finding generalized equivalence transformations for differential equations involving arbitrary functions

A new efficient method for finding generalized equivalence transformations for a class of differential equation systems via its related extended classical symmetries is presented. This technique can be further adapted to find the equivalence transformations for the mathematical model. It applies to classes of differential systems whose arbitrary functions involve all equations’ independent variables. As a consequence, any symbolic manipulation program designed to find classical Lie symmetries can also be used to determine generalized equivalence transformations and equivalence transformations, respectively, without any modification of the program. The method has been implemented as the maple routine gendefget and is based on the maple package desolv(by Carminati and Vu). The nonlinear stationary heat conduction parameter identification problem is considered as an example.     

Uniform controllable sets of left-invariant vector fields on compact Lie groups

A set of vector fields on a differentiable manifold M is said to be uniformly completely controllable (u.c.c.) if there exists a nonnegative integer N such that evert pair (p, q) of point of M can be joined by a trajectory, or positive orbit, of which involves at most N switches.

In this article we show that if M is a Lie group G and a set of left-invariant vector fields on G, N must be greater than or equal to dim(G)-1. We also construct sets of vector fields which are uniformly completely controllable in dim(G)-1 switches when G is the Lie group of any compact real form of g and g runs over all classical simple Lie algebras over .     

Dynamics and Control of Elastic Joint Manipulators

The investigations presented in this paper are based on our previousstudies where the modeling and control problems of rigid body manipulatorswere treated through the so-called vector-parametrization of the SO(3)group. The nice property of this parametrization, which also displays a Liegroup structure, is that it drastically simplifies some considerations andreduces the computational burden in solving direct kinematic problems,inverse kinematic problems and dynamic modeling by more than 30 hitherto.This statement, which is valid for models built through vector-parameter,becomes stronger in pure vector-parameter considerations. It is provedadditionally that the computational effectiveness of the vector-parameterapproach increases with the increasing number of the revolute degrees offreedom. Here we show that this can be used successfully in the problems ofelastic joint manipulators where, besides the real n links,nfictious links are included and an additional nrevolute degrees of freedom are involved. The present paper also considersthe role of group representations of the rotation motions in the modelingand control of manipulators with elastic joints. Dynamic modelsthrough vector-parameter and in purevector-parameter form are developed and the inverse dynamic problem isdiscussed. It is shown that the nonlinear equations of motion are globallylinearizable by smooth invertible coordinate transformation and nonlinear state feedback.