Constructing algebraic groups from their Lie algebras

A connected algebraic group in characteristic 0$0$ is uniquely determined by its Lie algebra. In this paper algorithms are given for constructing an algebraic group in characteristic 0$0$, given its Lie algebra. In particular, this leads to an algorithm for finding a maximal reductive subgroup and the unipotent radical of an algebraic group.     

New exact nonlinear filters with large Lie algebras

The ideas previously used (Stochastics. Vol. 5, pp. 65–92, 1981) to construct some finite-dimensional nonlinear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.In engineering language, the new filters are obtained from the old by smoothing the output through an n-pole linear system before adding observation noise in the usual way; this adds 2n to the Lie algebra dimension. In the simplest case (drift = tanh x, OBSERVATION = x) we put x through a one-pole described by a new variable ζ, and observe ζ + noise instead of x + noise; the new Lie algebra has additional generators ζ and ∂/∂ξ besides the four from the oscillator algebra, to give dimension 6.The filter, which gives a recursive construction of the conditional density, can be ‘derived’ by any of three (here equivalent methods: (i) direct integration of the Kallianpur-Striebel formula as a Gaussian integral; (ii) solution of a parabolic PDE with quadratic potential; and (iii) the Wei-Norman procedure.     

Distributions having bases which generate finite dimensional Lie algebras

Let X¹,…, X^k be real analytic vector fields on an n-dimensional manifold M, k < n, which are linearly independent at a point p ε M and which, together with their Lie products at p, span the tangent space TM_p. Then X¹,…, X^k form a local basis for a real analytic k-dimensional distribution x→D^k(x)=span{X¹(x),…,X^k(x)}. We study the question of when D^k admits a basis which generates a nilpotent, or solvable (or finite dimensional) Lie algebra. If this is the case the study of affine control systems, or partial differential operators, described via X¹,…, X^k can often be greatly simplified.     

On a new class of finite dimensional estimation algebras

We introduce a new class of solvable finite dimensional estimation algebras that contains both the linear and the Bene cases. The condition for finite dimensionality is expressed in terms of two matrices. They play an important role in understanding the structure of estimation algebras.     

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.     

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.     

Algorithms of Dynkin diagrams in Lie group machine learning

This paper uses the geometric method to describe Lie group Machine Learning （LML） based on the theoretical framework of LML, which gives the geometric algorithms of Dynkin diagrams in LML. It includes the basic conceptions of Dynkin diagrams in LML, the classification theorems of Dynkin diagrams in LML, the classification algorithm of Dynkin diagrams in LML and the verification of the classification algorithm with experimental results.     

Cascade decomposition of nonlinear systems, foliations and ideals of transitive Lie algebras

We relate the cascade decomposition of nonlinear systems to the joint notions of foliations and ideals of transitive Lie algebras.     

Parametrizing Del Pezzo surfaces of degree 8 using Lie algebras

For a Del Pezzo surface of degree 8 given over the rationals we decide whether there is a rational parametrization of the surface and construct one in the affirmative case. We define and use the Lie algebra of the surface to reach the aim.     

Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finite-dimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. Finally, we determine the maximal dimension of abelian subalgebras and ideals for non-decomposable solvable non-nilpotent Lie algebras of dimension 6 over both the fields ? and ?, showing the differences between these fields.     

基于稀疏字典的李群机器学习算法

李群机器学习理论被广泛应用于图像集分类中的数据表示和处理,并获得较优结果.由此,文中提出基于稀疏字典的李群机器学习算法.首先使用协方差矩阵对图像集建模,分析协方差矩阵构成的李群结构,应用对数映射将数据映射到线性空间中,得到数据的距离矩阵.再使用路标多维缩放对数据进行降维处理,降低运算成本.最后,使用带费舍尔判别字典学习进行分类.在YTC数据集上的实验证明文中算法具有良好的鲁棒性和准确率.     

Finite dimensional filters with non-linear drift IX Construction of finite dimensional estimation algebras of non-maximal rank

The idea of using estimation algebra to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It has proven to be an invaluable tool in the study of nonlinear filtering problem. In 1983, Brockett proposed to classify all finite-dimensional estimation algebras. In this paper, we give the construction of finite-dimensional estimation algebras of non-maximal rank. These non-maximal rank finite-dimensional estimation algebras play an important role in Brockett's classification problem.     

A note on estimation algebras on nonlinear filtering theory

The idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters. In his talk at the International Congress of Mathematicians in 1983, Brockett proposed to classify all finite dimensional estimation algebras. In this paper we consider some filtering systems. In a special filtering system: (1) We have some structure results. (2) For any arbitrary finite dimensional state space, under the condition that the drift term is a linear vector field plus a gradient vector field, we classify all finite dimensional estimation algebras with maximal rank. (3) We classify all finite dimensional estimation algebras with maximal rank if the dimension of the state space is less than or equal to three. A more general filtering system is considered. The above three results can be ‘used’ locally. Therefore from the algebraic point of view, we have now understood generically some finite dimensional filters.     

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.     

Controllability of right invariant systems on real simple Lie groups of typeF 4,G 2,C n,andB n *

We deal with controllability of right-invariant systems for some real simple Lie groups ofF ₄,G ₂,C _n , andB _n types. We prove that the so-calledcontrollability rank condition is a necessary and sufficient condition for controllability for an open class of systems. In other papers, analogous results were obtained for Lie groups of the remaining types (i.e.,E ₆,E ₇,E ₈,A _n , andD _n ) using a special property of the root systems of their Lie algebras.     

An estimation problem in compact Lie groups

In this paper an estimation problem is formulated and solved where the process evolves in a compact, semisimple Lie group. The problem is to find explicitly the conditional probability density of a random initial condition of a Brownian motion in the Lie group given a vector-valued function of the group-valued Brownian motion at some fixed positive time. This function is constant on orbits of the Lie group formed by conjugation. The conditional probability density function has a simple, explicit form using some orbit properties of the Lie group. Many physical problems can be modelled as evolving on compact, semisimple Lie groups, e.g. the motion of a rigid body with a fixed point.     

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。     

NML代数的滤子及其结构

首先在NML代数上引入MP-滤子与素滤子的概念,进而讨论了滤子和素滤子的基本性质,最后在全体素滤子之集上建立了拓扑结构。     

伪效应代数中的模糊滤子与模糊理想

在伪效应代数中提出了模糊滤子和模糊理想的概念,讨论了它们的性质;引入并研究了强模糊滤子和强模糊理想,得到了一些好的结论。