Lie Group Research Papers - Academia.edu (original) (raw)

This work studies the dynamic modeling method for a service robot with Omni-directional Mobile ManipulatorS configuration. Based on screw theory, Lie group notations, reciprocal product of twist and wrench, and Jourdain principle, the... more

This work studies the dynamic modeling method for a service robot with Omni-directional Mobile ManipulatorS configuration. Based on screw theory, Lie group notations, reciprocal product of twist and wrench, and Jourdain principle, the robot’s motion equations including the whole body manipulation are formulated with left invariant representation. A legible and canonical dynamic model representing the relation between the inputs and the generalized dynamic load wrenches is presented. Considering the tradeoff between the symbolic concision, the modularization in code realization and the computation load, the dynamic model is decomposed into succinct block factorizations, and the basic computation unites are boiled down to the adjoint map corresponding to each joint. The traditional Lie bracket operation is extended to a generalized form. Computation efficiency, for the coefficient matrixes of the system motion equation, is discussed based on its special representation form. The generalization of the modeling method with Lie group and algebra tool is also summarized.

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which... more

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L_1,...,L_n are homogeneous, we prove analogues of the Mihlin-H\"ormander and Marcinkiewicz multiplier theorems.

In this survey paper we consider some applications of the Wright function with special emphasis of its key role in the partial differential equations of fractional order. It was found that the Green function of the time-fractional... more

In this survey paper we consider some applications of the Wright function with special emphasis of its key role in the partial differential equations of fractional order. It was found that the Green function of the time-fractional diffusion-wave equation can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions.Finally, we discuss recent results about distribution of zeros of the Wright function, its order, type and indicator function.

We develop further basic tools in the theory of bounded continuous cohomology of locally compact groups; as such, this pa- per can be considered a sequel to (18), (39), and (11). In particular: - we establish sucien t conditions for a... more

We develop further basic tools in the theory of bounded continuous cohomology of locally compact groups; as such, this pa- per can be considered a sequel to (18), (39), and (11). In particular: - we establish sucien t conditions for a cohomology class of a discrete subgroup of a connected semisimple Lie group with nite center to be representable by a bounded dieren tial form on the quotient of the associated symmetric space by ; furthermore if : ! PU(q; 1) is any representation of any discrete subgroup of SU(p; 1), we give an explicit bounded dieren tial form on the quotient of complex hyperbolic space by which is a representative for the pullback via of the Kahler class of PU(q; 1); - if G;G0 are Lie groups of Hermitian type, we generalize to representations : ! G0 of lattices < G the invariant de- ned in (13) and we establish a Milnor{Wood type inequality. As an application: - we study maximal representations of a lattice in SU(1; 1) into PU(q; 1) for q 1, and - after establish...

This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2,R)SL(2,\R)SL(2,R) on the space of smooth functions on... more

This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2,R)SL(2,\R)SL(2,R) on the space of smooth functions on the Poincar\'e disc as a sub-representation of SL(2,R)SL(2,\R)SL(2,R) in the Weyl-Moyal star product algebra on R2\R^2R2. We indicate how it is possible to extend

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in (14) in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free... more

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in (14) in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrodinger equation and the central charge-free Virasoro algebra Vect(S1). We call sv the Schrodinger-Virasoro Lie algebra. We choose to present sv from a Newtonian

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A... more

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge

« La Physique mathématique, en incorporant à sa base la notion de groupe, marque la suprématie rationnelle…Chaque géométrie-et sans doute plus généralement chaque organisation mathématique de l'expérience-est caractérisée par un groupe... more

« La Physique mathématique, en incorporant à sa base la notion de groupe, marque la suprématie rationnelle…Chaque géométrie-et sans doute plus généralement chaque organisation mathématique de l'expérience-est caractérisée par un groupe spécial de transformations…. Le groupe apporte la preuve d'une mathématique fermée sur elle-même. Sa découverte clôt l'ère des conventions, plus ou moins indépendantes, plus ou moins cohérentes »-Gaston Bachelard, Le nouvel esprit scientifique, 1934 Tout mathématicien sait qu'il est impossible de comprendre un cours élémentaire en thermodynamique.

L’objet de l’exposé concerne les structures géométriques élémentaires de l’apprentissage machine, fondées sur le « gradient naturel » de la « Géométrie de l’Information », qui rend invariant par changement de paramétrisation le gradient... more

L’objet de l’exposé concerne les structures géométriques élémentaires de l’apprentissage machine, fondées sur le « gradient naturel » de la « Géométrie de l’Information », qui rend invariant par changement de paramétrisation le gradient d’apprentissage dans les réseaux de neurones par le biais de la matrice de Fisher. Après cette introduction exposant le « gradient naturel » [1] et ses extensions récentes aux réseaux de neurones profonds [2], nous développons de nouvelles mé-thodes pour étendre l’approche à des espaces plus abstraits et en particulier les groupes de Lie (matriciels).
L’apprentissage profond a été étendu récemment avec succès aux graphes, mais le thème émergent « (Matrix) Lie Group Machine Learning » [14][18][15][16][20][21] est une extension particulièrement intéressante pour les applications indus-trielles : reconnaissance de mouvements/cinématiques (série temporelle d’éléments du groupes SE(3)), reconnaissances de postures/gestes articulés [3](série temporelle de vecteurs d’éléments du groupe SO(3)), reconnaissance micro-Doppler [12](série temporelle d’éléments du groupe SU(1,1)) et en robotique (éléments de sous-groupes du groupe affine Aff(n)).
Nous exposerons l’extension de la notion de métrique de Fisher par le mathématicien Jean-Louis Koszul [4][6][17] sur les cônes convexes saillants. Pour l’extension de l’apprentissage machine aux groupes de Lie, nous présenterons les outils issues de la physique statistique à travers le modèle du physicien Jean-Marie Souriau de la « Thermodynamique des groupes de Lie » [7][8][19][22] basé sur la géométrie symplectique (application moment, 2 forme KKS « Kirillov-Kostant-Souriau » dans le cas non-équivariant [5], le cocycle symplectique de Souriau, les méthodes des orbites coadjointes [9] issues de la théorie des représentations de Kirillov des groupes de Lie [10][11]).
Nous terminerons par une illustration d’apprentissage machine pour les exemples canoniques de groupes de Lie matri-ciels classiques tels que les groupes SU(1,1)[13][23] (cas équivariant à cohomologie nulle), et le groupe SE(3) (cas non-équivariant à cohomologie non-nulle)[13].

We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the... more

We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the "transformation" of the commutative algebra of smooth functions on M in a new non-commutative algebra. These ideas lead in a natural way to Quantum Groups as deformation (or quantization,

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard... more

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.

We develop the "triangulated" version of loop quantum cosmology, recently introduced in the literature. We focus on the "dipole" cosmology, where space is a three-sphere and the triangulation is formed by two tetrahedra. We show that the... more

We develop the "triangulated" version of loop quantum cosmology, recently introduced in the literature. We focus on the "dipole" cosmology, where space is a three-sphere and the triangulation is formed by two tetrahedra. We show that the discrete fiducial connection has a simple and appealing geometrical interpretation and we correct the ansatz on the relation between the model variables and the Friedmann-Robertson-Walker scale factor. The modified ansatz leads to the convergence of the Hamiltonian constraint to the continuum one. We then ask which degrees of freedom are captured by this model. We show that the model is rich enough to describe the (anisotropic) Bianchi IX Universe, and give the explicit relation between the Bianchi IX variables and the variables of the model. We discuss the possibility of using this path in order to define the quantization of the Bianchi IX Universe. The model contains more degrees of freedom than Bianchi IX, and therefore captures some inhomogeneous degrees of freedom as well. Inhomogeneous degrees of freedom can be expanded in representations of the SU(2) Bianchi IX isometry group, and the dipole model captures the lowest integer representation of these, connected to hyper-spherical harmonic of angular momentum j=1.

We have analyzed, calculated and extended the modification of Maxwell's equations in a complex Minkowski metric, M4 in a C2 space using the SU2 gauge, SL(2,c) and other gauge groups, such as SUn for n>2 expanding the U1 gauge theories of... more

We have analyzed, calculated and extended the modification of Maxwell's equations in a complex Minkowski metric, M4 in a C2 space using the SU2 gauge, SL(2,c) and other gauge groups, such as SUn for n>2 expanding the U1 gauge theories of Weyl. This work yields additional predictions beyond the electroweak unification scheme. Some of these are: 1) modified gauge invariant conditions, 2) short range non-Abelian force terms and Abelian long range force terms in Maxwell's equations, 3) finite but small rest of the photon, and 4) a magnetic monopole like term and 5) longitudinal as well as transverse magnetic and electromagnetic field components in a complex Minkowski metric M4 in a C4 space.

We examine arguments in favour of tachyons and extensions of the Lorentz Group to include superluminal boosts. We offer support for the existence of tachyons from a recent finding in classical electromagnetic theory. With a slight... more

We examine arguments in favour of tachyons and extensions of the Lorentz Group to include superluminal boosts. We offer support for the existence of tachyons from a recent finding in classical electromagnetic theory. With a slight modification to the standard theory we observe that a single superluminal charge will bind to itself in a self-sustaining circular orbit, suggestive of a possible (modified) electromagnetic interpretation of hadrons. Symmetries in that theory are used in the subsequent analysis as a starting point in the search for extensions of the Lorentz Group. We discuss the issue of real versus imaginary coordinates in superluminal transformations of frame, and point out that the Lie group extended to include superluminal boosts is non-connected. We close with some speculation on the implications for faster-than light travel.