scholarly journals Lie-group methods

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 215-365 ◽  
Author(s):  
Arieh Iserles ◽  
Hans Z. Munthe-Kaas ◽  
Syvert P. Nørsett ◽  
Antonella Zanna
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Group Structure ◽  
Practical Interest ◽  
The Novel ◽  
Group Methods ◽  
Novel Theory ◽  
Numerical Integrators

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

scholarly journals Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Z. E. Musielak ◽  
N. Davachi ◽  
M. Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Algebraic Structure ◽  
Second Order ◽  
Lagrangian Formalism ◽  
Group Approach ◽  
Second Order Differential Equations ◽  
Gauge Functions ◽  
The Relationship

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS

2005 ◽  
Vol 02 (01) ◽  
pp. 111-125 ◽  
Author(s):  
PAOLO ANIELLO
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Similarity Transformation ◽  
Perturbative Expansion ◽  
Analytic Perturbation ◽  
The Matrix ◽  
Matrix Formula

We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

scholarly journals Lie Group Methods in Blind Signal Processing

Sensors ◽  
2020 ◽  
Vol 20 (2) ◽  
pp. 440 ◽  
Author(s):  
Dariusz Mika ◽  
Jerzy Jozwik
Keyword(s):  
Signal Processing ◽  
Lie Groups ◽  
Lie Group ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Search Space ◽  
Optimization Techniques ◽  
Blind Signal Processing ◽  
Blind Signal ◽  
Group Methods

This paper deals with the use of Lie group methods to solve optimization problems in blind signal processing (BSP), including Independent Component Analysis (ICA) and Independent Subspace Analysis (ISA). The paper presents the theoretical fundamentals of Lie groups and Lie algebra, the geometry of problems in BSP as well as the basic ideas of optimization techniques based on Lie groups. Optimization algorithms based on the properties of Lie groups are characterized by the fact that during optimization motion, they ensure permanent bonding with a search space. This property is extremely significant in terms of the stability and dynamics of optimization algorithms. The specific geometry of problems such as ICA and ISA along with the search space homogeneity enable the use of optimization techniques based on the properties of the Lie groups O ( n ) and S O ( n ) . An interesting idea is that of optimization motion in one-parameter commutative subalgebras and toral subalgebras that ensure low computational complexity and high-speed algorithms.

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

Dynamical Equations of Flexible Multibody Systems under Lie Group Framework

2015 ◽  
Vol 723 ◽  
pp. 215-223
Author(s):  
Wen Jie Yu ◽  
Zhen Kuan Pan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Multibody Systems ◽  
Group Structure ◽  
Translational Motion ◽  
Flexible Multibody Systems ◽  
Lagrange Equations ◽  
Flexible Multibody ◽  
Deformable Bodies

A new type of dynamic equations of flexible multibody systems is derived via virtual work principle together with floating frame approach. The absolute Cartesian coordinates are used to describe positions of floating frames’ origins of deformable bodies; The orientation transform matrices as special orthogonal groups are used to describe rotation motions; The modal coordinates are used to express small deflections of deformable bodies with respect to the corresponding floating frames. The resulting equations are mixed classic Euler-Lagrange equations of translational motion of bodies and deformation and Euler-Poinaré equations of rotation in Lie groups and Lie algebras, the Lie-Poisson equations as reconstruction equations obtaining Lie Groups from Lie algebras, constraint equations Lie groups and Lie algebras. In order to simplify the implementation, we separate the constant coefficient matrices in generalized mass matrices and generalized forces in detail. The results can be easily used to design geometric integrators of Lie group structure preserving.

scholarly journals On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations

2000 ◽  
Vol 3 ◽  
pp. 44-75
Author(s):  
Arieh Iserles ◽  
Antonella Zanna
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Homogeneous Spaces ◽  
Practical Interest ◽  
Exponential Map ◽  
Graded Lie Algebras ◽  
Graded Algebras ◽  
Algebraic Setting ◽  
Cayley Transformation

AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

scholarly journals LIE GROUPS AND NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS: INVARIANT DISCRETIZATION VERSUS DIFFERENTIAL APPROXIMATION

2013 ◽  
Vol 53 (5) ◽  
pp. 438-443 ◽  
Author(s):  
Decio Levi ◽  
Pavel Winternitz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Group Theory ◽  
Differential Equations ◽  
Numerical Solution ◽  
Lie Groups ◽  
Lie Group ◽  
Numerical Solutions ◽  
Differential Approximation ◽  
Lie Group Theory

We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the “first differential approximation” is invariant under the same Lie group as the original ordinary differential equation.

The geometry and invariance properties for certain classes of metrics with neutral signature

2016 ◽  
Vol 13 (06) ◽  
pp. 1650080 ◽  
Author(s):  
Jean J. H. Bashingwa ◽  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Algebras ◽  
Lie Group ◽  
Partial Differential ◽  
Invariance Properties ◽  
Group Methods ◽  
Variational Symmetries

In this paper, we study anti-self dual manifolds endowed with metrics of neutral signature. Since the metrics depend on solutions of, in some cases, well-known partial differential equations (PDEs), we determine exact solutions using Lie group methods. This concludes specific forms of the metrics. We then determine the isometries and the variational symmetries of the underlying metrics and corresponding Euler–Lagrange (geodesic) equations, respectively, and establish relationships between the resultant Lie algebras. In some cases, we construct conservation laws via these symmetries or the “multiplier approach”.

scholarly journals Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 17-29 ◽  
Author(s):  
Kenneth D. Johnson
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Nilpotent Subgroup ◽  
Iwasawa Decomposition ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Simply Connected

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

Hom-Lie group and hom-Lie algebra from Lie group and Lie algebra perspective

2021 ◽  
pp. 2150068
Author(s):  
S. Merati ◽  
M. R. Farhangdoost
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Group Structure ◽  
Point Of View ◽  
Lie Group Theory

A hom-Lie group structure is a smooth group-like multiplication on a manifold, where the structure is twisted by a isomorphism. The notion of hom-Lie group was introduced by Jiang et al. as integration of hom-Lie algebra. In this paper we want to study hom-Lie group and hom-Lie algebra from the Lie group’s point of view. We show that some of important hom-Lie group issues are equal to similar types in Lie groups and then many of these issues can be studied by Lie group theory.

Sign in / Sign up

Export Citation Format

Share Document